of Mr. Cayley and Mr. Cockle. 459 



iv. "Combining (e^, (E^, we have 



in which (i b , fi 4 , . . /jl must all of them, in accordance with what 

 has been stated in art. i., be expressible as rational functions of 



"But as (U), the equation on which u depends, is rationally- 

 irreducible below the sixth degree, the coefficients 



of the equation (e , 1 ) ought severally to vanish. 



" When, therefore, u, v are equal to a, a 5 respectively, (tfj) 

 ought to become 



v = u 5 , 



which is precisely identical in form with the equation (Ej) itself. 



"We thus perceive that we cannot derive any aid from 

 Lagrange's theory in expressing a 5 as a rational function of a 

 and of Ap A 2 , . . A 5 . It remains to show that (e 2 ) also is inap- 

 plicable when u=a, v = a 5 . 



" The equation (e 2 ) is supposed to be obtained directly from 

 the theory of Lagrange. But an equation identical in its ulti- 

 mate form with (e 2 ) ought to be found by combining (ej and (U). 

 Whence 



if 



u=a, v=a 5 , 



since (e } ) here becomes, or at least ought to become, a 5 =a s . 



"Of the interpretation of (e v U)', which is indicative of an 

 incongruity, I intend to speak in another place." 



From this extract it will be perceived that (e/) expresses, by 

 means of mathematical symbols, the exceptional nature of the 

 homogeneity existing between a and a 5 . 



3. How then does Mr. Cayley answer the question, " Is La- 

 grange's theory inapplicable to the case of u-=-ol, v=ot, b , in virtue 

 of the equation (e'j) V 3 



" But if," remarks Mr. Cayley — shunning the true point in 

 dispute, and not giving due heed, as we shall see, to the defi- 

 nition of the word objection, — u but if my objection be (curiously 

 or otherwise) irrelevant, then the proposition I contend for might 

 be admitted without prejudice to Mr. Jerrard's results : this 

 proposition is, that Lagrange's theory is applicable to the case of 

 u=.a, v = a, 5 , which is a case not excluded by the only exception 

 (the case of equal roots) to the general theory, and therefore, 

 notwithstanding the equation (e'j) or anything else whatever, 

 coming within the general theory. Mr. Jerrard, in his reply, 



2H2 



