570 Mr. G. B. Jerrard's Befledions on the ResoltUion 



thence arise by ^H, we shall, on taking ^ successively equal to 

 0, 1, 2, 3, 4, obtain the five functions ^S, ^S, gSj 3H, 4E ; the 

 first of which, qH, will be equal to &\ f + 0'^2 / + ©"^2 / 



We may therefore form an equation of five dimensions, 



(S-oH)(E-,S)..(H-4H) = 0; . . . (ab.) 

 the coefficients of which, when arranged according to the 

 powers of S, shall be rational functions of Py/^j\. For, in 



consequence of the symmetric manner in which a' and a" will 

 enter into these coefficients, the symbol %/ ^ as well as .y^, will 

 disappear from the calculus*. 



41. Now 0'^i^y + . . + 0"^i,/ is» as we have already stated, 

 a rational function of Pyj and cannot therefore, when the roots 

 a?!, ^2J • • -^5 change places among themselves, receive more 

 than twelve different values. If, indeed, we consider that 



(0'^,/+ .. +0'V)(^.r)(lO="(®"i'/+-- +®"'i./Kr.O. 



while P///3y\/St\ is not equal to ^fuyy but to P/'(y?); we 

 shall instantly perceive that 0'^,/+ .. + 0"^i^ymust admit 

 of becoming a root of a determinate equation of the — ^^th 

 degree expressible by 



* We see at once that we are permitted to suppose 



■ o3 - ^i («'» «") + ^2 K «") + 4'2 («> «') + fi («"» «'), 



oS will therefore belong to the class of expressions 



o^F(a',«")+o^K«'). 

 There may also subsist 



jS = l^^, («', «") + ^4., («', «") + <'^2 («", «') + ^^1 («", «')> I 

 4H == ^4/1 (a', «") + '*4'2 («', «"} + '4^2 («". «') + '^1 («"' *')• J 



and 



,H =^^|/i(«',a") +^4'2K*") +'4'2(«'U') +'^i(«'>'),l 



3B = s^p, («', «") + "4/2 («'» '^'O +'4^2 ('^"^ '^O + '^1 (^"' «') ' J 



as is evident. If, therefore, we suppose ^E and gE to be de- 

 noted by 



,^(a',«"), 2^(«',a") 



respectively, 4S and 3E must take the forms 



Whence it appears that «' and a" are involved symmetri- 

 cally in the equation (a b.). 



There are some other results connected with the equation 

 (a b.) which I should like to verify, but I have no time to do 

 so now. 



