## Uniplanar Algebra: Being Part I of a Prop©¡deutic to the Higher Mathematical Analysis |

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... Properties of hyperbolic ratios 55 48. Geometrical construction for hyperbolic

ratios 55 49. Agenda : Properties of the equilateral hyperbola .... 58 50. The

Gudermannian 58 51. Agenda : Gudermannian formula 58 52. Proof that limit (

... Properties of hyperbolic ratios 55 48. Geometrical construction for hyperbolic

ratios 55 49. Agenda : Properties of the equilateral hyperbola .... 58 50. The

Gudermannian 58 51. Agenda : Gudermannian formula 58 52. Proof that limit (

**sinh**«.)/ ... 55 ÆäÀÌÁö

An important class of exponentials, Whic-h because of their relation to the

equilateral hyperbola are called the hyperbolic s-i-ne,-cosine, tangent, Cotangent

, secant and cosecant, and are Symbolized by the abbreviations

Coth ...

An important class of exponentials, Whic-h because of their relation to the

equilateral hyperbola are called the hyperbolic s-i-ne,-cosine, tangent, Cotangent

, secant and cosecant, and are Symbolized by the abbreviations

**sinh**, cosh, tanh;Coth ...

57 ÆäÀÌÁö

and by virtue of the relations cosh* u — sinh2 u = 1 , and x* / a' —y I a' = 1, y - =

s1nh u. a Also, since x* — y* = a* and x* — NQ* — a2, therefore NQ = NP, and

we have x v - = sec 6 = cosh u, - — tan 6 =

a ...

and by virtue of the relations cosh* u — sinh2 u = 1 , and x* / a' —y I a' = 1, y - =

s1nh u. a Also, since x* — y* = a* and x* — NQ* — a2, therefore NQ = NP, and

we have x v - = sec 6 = cosh u, - — tan 6 =

**sinh**u, and the co-ordinates of Q beinga ...

58 ÆäÀÌÁö

A straight line through P and Q passes through the left vertex of the hyperbola

and is parallel to OV. (5) . The angle APN= one-half the angle QOIST. 50. The

Gudermannian. When 6 is defined as a function of u by the relation tan 6 =

(Art.

A straight line through P and Q passes through the left vertex of the hyperbola

and is parallel to OV. (5) . The angle APN= one-half the angle QOIST. 50. The

Gudermannian. When 6 is defined as a function of u by the relation tan 6 =

**sinh**u(Art.

59 ÆäÀÌÁö

To Prove Limit [(

, during the interval of time /' — t, P moves over the distance x' — x, Q over the

distance u' — u. Then speed being expressed as the ratio of distance passed

over ...

To Prove Limit [(

**sinh**u) /u] = 1,whenaio. In the construction of Art. 23 suppose that, during the interval of time /' — t, P moves over the distance x' — x, Q over the

distance u' — u. Then speed being expressed as the ratio of distance passed

over ...

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