148 Mr. J. Cockle on a Theory of Transcendental Roots. 



cess. Thus far I have used the non-symmetric form ; but inas- 

 much as the symmetric one leads to 



# = <£<(, a=^<f>x = ^xf)a J 



we have, in </> 2 a = a, a relation which further limits the form of (f>. 

 Whichever form we employ, the first derived equation may be 

 depressed to the second degree in #, and for the latter we find 



_ ,dx dx ; _ , - 



x n ~ ! -j a -r: x + a n ~ l =0, 



da da 



or, reducing, 



In treating of equations which involve more than one para* 

 meter, we must employ the differential equations 



and however numerous be the symbols in the equation 



flv,w,...y,z) = 0, 



we shall, provided that relation be symmetric, in the end be led 

 to relations of the form 



z=^y, y=^z=y*y. 



I mention this because it may be found desirable to study 

 some canonical forms other than those pointed out by Mr. 

 Jerrard. When a is greater than unity, logarithmic may replace 

 the above circular functions, and, when the parameters are inde- 

 pendent, the 8 of the calculus of variations may replace d, and 

 we shall have the identical equalities 



#(«,i,..)=o, 8/^(a,*,..)=o, sy<K«,A,..)=o,... 



I noticed the auxiliary condition ^a — a in a footnote at p. 133 

 of this Journal for February 1846 (S. 3. vol. xxviii.). Subsi- 

 diary conditions are necessary, even in the foregoing discussion 

 of a cubic, for suggesting or completely ascertaining the form of 

 <f) ; and it will be observed that 



dx . . <//C+sin -1 «\ 



where C is a constant. 



4 Pump Court, Temple, London, 

 June 30, 1860. 



