370 Mr. J. Cockle on Transcendental Roots. 



we deduce by differentiation 



d 2 x __ a<pb 1 d<\>b db 

 da* "■""" W~"3b"W' da 



a<l>b I ( <l>a\/ a\ 

 " 3b 3 36 V 3a) \ b)' 



This becomes, on reduction, 



d 2 x a dx 



+ O.M ^ =0 f 



*fo 2 1-a* da ' 3 2 (l-a 2 ) 



a linear differential equation, the integral of which is 



A . /sin- 1 a j\ 

 #=Asml — ■= — + i> ). 



Substitute this value of x in the given cubic. The result must 

 vanish identically and independently of a ; and from the relations 



A 3 -4A=0, A 3 -8 = 0, sin3B=0, 

 we infer 



rt . /sin -1 « + 2w7r\ 



a?=2sln V 3~ -J' 



The unsymmetric trinomial form 



x n —nx + n{n— 1)« = 



is in some respects more convenient than that employed in the 

 " Sketch," and, through 



„ dx x dfx * 

 J da n da 

 it leads to 



Each unsymmetric form indeed leads to a derived equation, 

 which, by properly combining it with the given equation, may 

 be made linear in x. 



I might have supported an early objection (S. 3. vol. xxxv. 

 p. 436) which I took to a portion of Mr. Jerrard's investigations, 

 by citing a paper* by the late Thomas White, of Dumfries Ma- 

 thematical Academy, which (received April 1816) was printed in 



* " On the Algebrai cal E xpansion of Quantity, by Division and Evolution; 

 and on the Symbol V — 1, which is usually considered to denote impossible 

 or imaginary Quantity." White (p. 61), in reference to the series " usually 

 but whimsically .... denominated neutral" says that " all ambiguity va- 

 nishes when it is considered that this series is finite, and that attention 

 must always be paid to the remainder." 



