374 Sir William Rowan Hamilton on Equations of the Fijlh Degree. 



v"\„ = H„ + \/h., + a/his + A/H,e + \/h„ - ^/H,8 ; (a") 



534 



V'"435 = Hj3 - /h,, + ^H,, - •/h,6 + a/h,, + ^H,8 ; (b") 



and they give, as the simplest of the expressions deduced from them, the two 

 following, which are analogous to that marked (220) : 



H. = i (V'^340 + ^'\. + V'\34 + V'^M + V'^«3 + V'\35) ; (234) 



H,a = i (^"^345 + V"»4« + ^"^534 + v"^354 + V'^^a + ^'\^)- (235) 



For the case of the equation (147), and the arrangement of roots (148), we find 

 the numerical values : 



I v'3« = - 126b - 7c ; I v'453 = 202b - 11c ; f y\^ = 25b + 50c ; | 

 f v",3, = - 126c + 7b ; f v"^3 = 202c + 11b ; f v"3^ = 25c - 50b ; J ^ 



fv',35=-18B + 47c; fv'«3= 100b -175c; f v'3^ = -61b- 52c; ] 

 fv"3«=-18c-47B; fv%=100c + 175b; fv",3,= - 61c + 52b; i ^ ^ 



which may be obtained, either by the method of article 43., combined with the 

 values (221) (222) of the twenty-four functions x; or by the formulae (210), 

 combined with the following table : 



fT,3,3= -175b-25c; |t^35=-150-11b-77c; 



|t,453= +377b + 89c; fT^3 = 450 + 111b + 27c + 200d; • (238) 



f T,53, = 150 + 77b - 11c ; f t^ = - 450 - 111b - 27c - 200d ; . 



and with the condition, that, if we write for abridgment, 



Ticde = T^°'jcde + Bt'jc* + CT" tcde + I>T'"4cde, (239) 



we have in general the relations, 



Tedcb = T^°^6cde — BT'jcde — CT"jc&, + 'DT"'icdt ', j , 



And hence, for the same equation of the fifth degree, and the same arrangement 

 of the roots, we find, by (54) and (55) : 



H, = - 2-* 3-' 5* (10975 + 706d) ; 

 H,3= - 2-* 3- 5* (10975 - 706d). 



I (241) 



