﻿900 A Kinematic Solution of Equations. 



unity, but the linkage resulting will be of the second sort, 

 and will be indicated by the equation : 



O={3acos0+acos30} — {36 sin 0-& sin 30}. 



In this case tan 6— — , tan 2 0= — ^ when 6 has been found 

 by the linkage. a?1 lC ' 2 



The trisection of the area contained between the hyperbola 

 whose seraiaxes are a and b, its semiaxis a, and a radius 

 vector drawn from its centre to a point whose abscissa is «r 3 , 

 by other straight lines drawn through the centre, may be 

 effected by a linkage. 



Let x x and x 2 be the abscissae of the points of the curve 

 required by the problem, x x being smaller than x 2 . 



The area of such a figure is equal to the quantity — ^- . 0, 



in which is a quantity such that 1 r = acosh 0, i/ = b sinh 0. 

 The problem, so far as x x is concerned, therefore amounts 



to finding cosh when cosh 30 is given, for then — =cosh 0, 



x« ^ 



when — = cosh 30. Cosh 30 can be expressed in powers 



of cosh 0, thus : 4 cosh 3 — 3 cosh 0. 



Now cosh can be always represented by - . Hence 



cos V 



# 3 4 3 



a cos 3 6 cos 6 ' 



° r 4-3 cos 2 0-*± cos 3 0=0; 



a 



whence in cosines of multiples of 6 



0=10-3- 3 cos 0-6 cos 26- - 3 cos 30. 

 a a 



This formula indicates the linkage to be employed to find 

 cos 6. It is of the first sort. When this cosine is deter- 

 mined, #! = ^-£ , .r 2 is found directly from^ from the equations 



u\ 2 = a cosh 20, lV] — a cosh 0, 

 from which 2 l r l 2 — a 2 



p 2 = 



