the purpose of ascertaining whether they are dishonest or I am quite sure that in that court (the Insolvent Court) researching inquiry is known and practised, it is found essary to be applied to every case as the means of disag its true character and merits." TURES, METHOD OF, P. C., is given the mode to which we must have recourse, in order to find particular values of Q, when the general methods for determining it fail. In this article we contine ourselves to what is most useful in operation, as a summary for the advanced student, not an explanation for the Namelessness must not be presumed: faultiness is to be learner. Properly speaking, the problem requires some med: it may or may not be that which is told by the addition to make it definite. Thus 2x has a for a primitive rd fraud; the precise shade cannot be presumed; the cha-function, and also +C, C being any constant quantity and degree are to be learned through a deliberate and whatever. In the present article, we shall neglect this conred inquiry. It is misrepresentation to say that fraud is stant altogether, reminding the reader that he must never ed and punished on presumption; the coercion which omit it in any application. If he should find in different Be purely punishment is now necessary coercion to the books different functions given as the primitives of one and ization of a question in which presumption is and ought the same function, he will always find that those different bagainst the party coerced. The debtor in execution is primitives differ only by a constant quantity. Thus (1-x)applicant for indulgence; he has to establish his case; and (1-x)-1 both occur as the primitive of (1-x)2; but ne is at liberty to institute proceedings towards this ques- they only differ by a constant, namely 1. instantly on his arrest; and not only is he at liberty to texemption from the consequences of the injury which he done to the particular party who has pursued him, but to the same opportunity for acquiring a privilege against ry person in the kingdom towards whom he stands in a alar predicament: on giving to the true owners a part of e property, or on showing that there remains no part to sier, he receives, if excuse is found for granting it, this mat Leon-a total freedom for the future of person and prorare that if ever he become in the full and fair sense ewards of ability to pay, there will reside in a competent the power to ascertain that ability and to exact that mest. It is almost unnecessary to say that these results ought to be enjoyed without that fall disclosure of the history Es property which is found in the schedule of an insolvent that full opportunity for the creditors to challenge history; and that fair, deliberate, and effective investi. wa of its truth which is made in that court.' These general arguments in favour of the justice of final serien are supported by Mr. Law with facts equally g. which also prove the efficacy of such arrest. The de in which he has examined the arguments in favour of shing arrest, which are derived from certain returns, is ely convincing. The efficacy of arrest must not be by the extent of dividends made in the Insolvent Court, or the proportion of unfavourable judg though it must be remembered that the dividends none at all, as some people suppose. 1 In the common process of integration, the actual passage from the differential coefficient to the primitive is always an act of memory. The algebraical work which occurs is always used either to reduce a form in which memory will not serve into one in which it will, or else to reduce the given differential coefficient to two terms, one of which can be integrated by memory, and the other of which is more simple than the original quantity. The functions in which the simple remembrance of the forms of the differential calculus is of use are as follows:xn+1 fde=x, fadx=ax, fx”dx=" dx = To these should be added the following, which may be obtained in various ways from the methods of this article, or from peculiar artifices which are found in works on the subject. dx √(x*±a2) -dx a x 1 Cos a =log {x±√(x2—a°)} √ (a2 — x2)2 de is clearly shown by Mr. Law that arrest does make ray, who do not pay till they are arrested; it is found the examination to which insolvents are subjected ex&great amount of fraud; and it is also certain that the eer of those who are induced to pay by the fear of arrest nsiderable, just as the fear of other punishment prevents persons from committing crimes, who have no other tre to deter them. The fear of arrest is precisely that derating weight which is wanted to induce those whose esty is wavering to incline to the right side. The arguments of Mr. Law should be read by every man wishes to form a sound judgment on the law of insolvent Mrs in England; and so much of his arguments as have e been given, may help to diffuse some juster opinions on et in which a sympathy with debtors, to the total forbess of creditors, has led many well-meaning people to t conclusions that tend to unsettle all the relations of ty, and to confound honest men and rogues. Some valuobservations on the laws relating to Imprisonment for Je by Mr. Commissioner Fane are printed in the Banker's azine, No. xix., October 1845. He concludes 1st, the remedy given to creditors by the seizure of goods 1. The reduction of such a form as fXdr to another form a fieri facias [FIERI FACTAS, P. C.) is a delusive re-fVdv, in which v is a different variable. Thus f(a+o)”xdx *; and 2nd, That such remedy, instead of being bene- can be immediately reduced to }ƒ (a2+x2)nd.(a2+x2) or to the creditor, whom it is intended and supposed to do, where v means a2+. The second form is immeactually prejudices him, by enabling the debtor more tally to cheat him under the form of law; and, there-diately seen to be integrable. Cases of this kind are so varifar as relates to this branch of the subject, the power and recognising them at sight. Sometimes a slight transous that the student must form the habit of looking for them, h each ereditor now possesses of seizing his debtor's under a fieri facias for his own exclusive benefit, is a formation is required, thus; (1-+ae)'dr, when reduced to ous power which ought to be abolished. '(ε ̄*+a) ̄ ̄1ɛ ̄ˆdx clearly shows the form -v'de, where v INSURANCE, MARINE. [MARINE INSURANCE, P.C.] is e ̃ˆ+~. INTEGRATION. In the article INTEGRAL CALCULUS, PC., the meaning of an integral was explained. The prearticle is devoted to the operation of integration, that is, ing the primitive function which has a given function for erential coefficient. Having given P a function of a rethat dQ: de may be P. In the article QUADRAP. C. S., No. 97. ftanx dr=-log cos x fcotr dr=log sin r. Among the peculiar artifices of integration may be reckoned the following, which are perhaps nearly all that can be useful to a learner : 11. There are several cases in which the following sion of the theorem known by the name of John B may be useful. Let u', u', &e. be the successive diffe coefficients of u with respect to x, and let 1, V2, V3, the successive integrals of v with respect to r: then fudv=uv―u'v1+u'v2¬u'"v3+ · ± u(")v2 #su3~} This is particularly useful when a is a rational and int function, and v is successively integrable with ease, a u is ɛɛ, sin ax, or cos ar. The process can then be coati until the remainder vanishes. 12. In the case of order, where pr and r tional and integral functions, the integration is alway sible as soon as all the roots of 40 are found. The in FRACTIONS, DECOMPOSITION OF, P. C. S., must be plied. When this is done, and the function thereby to the sum of terms of the form A(x-a) ̃"dr, the inte tion gives no trouble. 13. In the case of a pair of irrational roots, a±ß each occurring once, the sum of the terms which they can be reduced to the form Л(Ax+B)dx: {(x−a)2+ß3} the first term of which can be integrated as in (1), leaving the integral of which is the second term, which can be simply integrated. 6. The process known by the name of integration by parts, consists in reducing the form Xdr into any convenient form Vdv, and using the obvious theorem 8. The use of the equation of reduction depends upon our being able at last to reduce the question to that of finding a visibly known integral. Thus, if in the preceding n be an integer, we must at last come to fxdx, or fedx, which is known. But if n were a fraction, no reduction of the value of n by units at a time would lead to an integrable form. 9. The integrable form at which we arrive by successive reductions is called the ultimate form. It frequently happens however that the reductions proceed by two or more steps at a time, in which case two or more ultimate forms result. For instance Vn=f(a2x2)-x"dx has for its equation of reduction x1−1 √ (a2 − x2), n−1 n n -aoV 2-2 Α 2 tan В B+Aa log{(x − a)2 +ẞ2 } +'? = β 14. When pxdx is a function of powers of any one ca ax+b, it can, if irrational, be reduced to a rationa by assuming ax+b=vm, where m is the least comrad tiple of all the denominators in the exponents. For d comes mvm-dv: a, and every power of ax+b bec integer power of v. 15. The function x (ax+b)"dx can be integratedy either m or n is a positive integer: when n is intege simple expansion; when m is integer, but not a, by ax+b=v, and substituting. But when both m and negative integers, let x=1:y and after substitution, a+by=w, and substitute for y. 16. The function rdx: (x"a") can be easily integ by decomposition of fractions, the denominator never equal roots. The same may be said if we subst x2+2ba"x"+a" in the denominator. 2n 17. In x'(a+bx)'dx we have an integrable fune whenever either of the following is a positive integer:- T r+1 -, or ---- -t r+1 S The substitutions which succeed in the two cases are a+brvo, and ax+bu being the denominator of t. 18. The following transformation involves a large ta of obvious cases, and is constantly occurring. If fød then fp(ax+b)dx=4(ax+b): a. Thus in no list would fcos(ax+b)dx be set down. fcosx dx has been given. 19. The following integrals are worth giving separately dr x √ (a2±x2) dx √(2ax-x2) 1 log a which comes under one or another of three previously g forms, according as b2-4ac is positive, nothing, or negat dr log(a+bx+cr2)-20 a+b+co √ (a+bx+cz!) = √ clog{2cx+b+√ { $c(a+br+cr°) 22. Let P stand for Ax"+Bx', (m, n) for fxmPd, for m+1+na and m+1+nb, and c for a-b. We then have h(m, n)+ncA(m+a, n−1)=x"+1pa g(m, n)-ncB(m+b, n−1)=x+1pa gA(m, n)+(h-c)B(m-c, n)=x"−a+1p+! hB(m, n)+(g+c)A(m+c, n)=xTM¬b+1pn+! the first pair of which formula of reduction can be found be it positive or negative, and for m from the second The most useful cases are those in which a=0, b=1, i in which a=0, b=2. 23. Let Va=ƒ(x2±a) ̄"dr. Then (2n-2)a2 (2n-2)a2 24. Let Va2)"dr. Then Vn-i. 5. Let Va—x2) ̄"dx or f(a2 -x2)"dr. The equa s of reduction are those in (23.) (using the + in ±) and (24), and writing a- for ± a2. 26. Let Vfx" (a2±2) "dr. Then -1 V11 == 2(n−1) (a2±x2)" -1 28. Instead of giving a large number of forms which are all derivable from (22.), it will be better to give an instance of the derivation in full. Let the case be fr" (2ax-x2)"dx, and let the formula be required to reduce both m and n in numerical magnitude. Here, to transform the formulæ in (22.) For m write-m; retain n. For a write 1; for b, 2. For g, -m+1+n; for h, -m+1+2n. For c write -1. The first formula connects (− m, n) and (—m—i, n − 1), the second, (m, n) and (−m −2 n − 1); the third, (—m, n) and (-m-1, n); the fourth, (m, n) and (−m+1, n). By either of the first two we can therefore reduce both; by either of the last two we can reduce m only. Observe that whenever a formula will serve to raise either exponent it will also serve to reduce it. Thus, if a formula were V„=4(m, x)+4m. Vm+1 write m-1 for m and we have, by transformation, (m −1, x) 4(m-1) (m-1) + 1. -V V -V, 1 (2ax-x2)" 2na 2n-m+1 m-1 2n-m+1 732-1, n-1 1 = m, n n-m+1 n V n-m+1'm-2, n−1. If it were required to reduce n in the preceding without '(2ar – xo)”, into the form altering m, throw the formula x "-" (2a-x)" and use the first of the four formulæ. 29. All the preceding forms involving "P", are particularly in use when n is a fraction, positive or negative, with the denominator 2. These in fact form the most usual Formulæ involving the powers of A+B+C are so little wanted, that they are better omitted in a work in which space is of importance. TAI " do m-1 n-1 m m-n m-n+2 n-l S S m-2 MR S do smde c7-2 We have now given most of the forms which will be useful in an ordinary work of reference. Further forms and examples will be found in many works on the integral calculus, but the largest collection is in Meier Hirsch's Integraltafeln,' Berlin, 1810, 4to., a work of which there is also an English edition. We have omitted notice of a great many such forms as fxnardx, fænεax cos nadr, &c. which are little used, except in particular cases. When paarde can be integrated, it follows that a cos bids, &c. can also be integrated, since the second can be made into the sum or difference of two functions of the first form, by putting for cos bx or sin bæ their exponential values. The question of the possibility of integration in finite terms can often be settled by the following theorem:-Integration and differentiation, with respect to different variables, are convertible operations; thus If therefore fudr can be found, so also can du: dy)dx, if y be not a function of r. From this it will be seen that whenever prade can be integrated so can predx, which is obtained by n differentiations with respect to d; and also that whenever px, xdx can be integrated, so can pr" (log x)dx, which is obtained by m differentiations with respect to n. Functions involving the transcendental forms sin-1px, &c. can sometimes be reduced to more algebraical forms by integration by parts. Thus, V sin X,desin-XVdx INTEGRATION, DEFINITE. In the preceding article we have given some idea of the usual modes of integration. The results, which in the present article are given under the name of definite integrals, are mostly cases in which it is possible to find an integral when both limits are given [INTEGRAL CALCULUS, P. C.]; but not possible to find the integral in all cases. If we can integrate orde generally, that is, if we can find the function 2, of which px is the differential coefficient, we can always express the integral, the limit of the summation in the article just referred to, as follows: greater number of integrals. Consequently the integral themselves become modes of expression, and frequently only ones. When we find a language with which we lan much to do, and which has words which cannot be transatl we adopt the words of that language into our own. cisely the same thing is done in the case of definite inter Thus in FACTORIALS, P. C. S., we adopt the integ Sadr, as the fundamental mode of expression fa function till then inexpressible, which becomes 1.2.3. whenever n is an integer, and remains intelligible, though very casily found, when n is a fraction, Further to illustrate this, let us suppose that the integr calculus had made some progress before the conception logarithm had been formed: a thing which might easily ha happened. It would then have been found that fzwholly unattainable, a function which algebra could not press in finite terms. It would therefore itself have bee a mode of expression, and it would soon have been pro that Here then would have been an obvious indication of existence of a, function proper to be made use of in perf ing multiplication by means of addition, &c.; and tables the values of fade would have been formed by the uth of quadratures [QUADRATURES, P. C.] or otherwise would, so it happens, have been a much easier task than which fell on the first calculators of logarithms. For all however it happens that we are prepared by knowing rithms and their properties; so that fa dx is seen to log. +C, and fxdx to be loga: the logarithms through this article being Naperian. But we are not equally. for fedt, or for fe ̄*x"dx (except when n is integer) cos xdx: and accordingly we are obliged to study properties of these functions as fundamental modes of exp sion. u =√ + * ¥ (x + 2v √at) ε ̄dv. where may be the symbol of any function. From this is clear that the given differential equation has number solutions which ordinary symbols are incapable of express in finite terms. The treatise in the Library of Useful Kno ledge on the Differential Calculus, Gregory's Examples the Differential Calculus,' and the Cambridge Mathematics Journal,' contain various examples of this mode of expressio applied to differential equations. We now proceed to give a selection from the enormog number of definite integrals which has been given. The have been found by detached methods, so that we could to attempt to give anything more than the results. Our articl is intended for reference to the forms which it is probable will be noted in future elementary works, and which the mathematical reader may also wish to refer to. In order to avoid risk of broken or dropped letters, in an article in which the correct printing of the limits is of the utmost in portance, we shall print what is usually denoted by pror in the following way, fordx [a, b]. Any conditions as to the values of constants will be expressed before the integral need hardly be said that the article FACTORIALS, P. C. S., | from these come rdx (m+l<n) /** [0, ∞ ] [0, ∞] = n sin cos bxdx [0, ∞ = 0, f sin bædx [0, ∞ ]=1; and from these come two equations which have been much used, long before they were openly expressed, sin 0, cos∞ = 0. Some difference of opinion exists about these equations, which in fact involve a great deal of what has been done r (m+1) г (n−m-1) by mathematicians in the last twenty years. Гл I'm. In г(m+n) a2 (1+a)" г (m+n) (ʼn positive) ƒ (—log x)”-1dx [0, 1]=rn (and a positive) fx-1 (—log x)”-1dx [0, 1]=m ̄"rn (a and a positive) -1dx [0, ∞ ] = a¬"In (a positive) ƒ e-*"dr [0, ∞ ]= 1 1 n n Tn.cos{n tan(b: a)} Tn. sinn tan-1(b: a)} = +1 a <b But when a integral is 1 n+ dx sin bx 1+x2 m [0, ∞] = 2m E 2mb±c, m being an integer, the preceding |