THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 215 
+ 34(66413) + 30(66350) + 24(66332) + 34(66314) 
+ 24( 66242) + 24(66233) + 12(66134) + 48(66125) 
+ 30(66053) + 30(65630) + 30(65603) + 34(65531) 
+ 30(65513) + 40(65441) + 18(65423) + 18(65414) 
+ 36(65405) + 30(65351) + 38( 65342) + 8(65324) 
+ 2(65243) + 46(65234) + 28(65225) + 34(65153) 
+ 36(65045) + 24(64622) + 36(64550) + 2(64532) 
+ 38(64523) + 40(64514) + 18(64451) + 18(64352) 
+ 16(64334) + 46(64525 ) + 8(64235) + 40(64154) 
+ 18(64145) + 12(63641) + 24(63632) + 24(63623) 
4.12(63614) + 46(63542) + 34(63515) +8(63452) 
+ 16(63443) + 2(63425) + 38(63254) + 18(63245) 
+ 30(63155) + 48(62651) + 48(62615) + 28(62552) 
+ 8(62543) + 18(62534) + 46(62453) + 38( 62435) 
+ 2(62354) + 34(61643) + 34(61634) + 18(61544) 
+ 30(61535) + 40(61445) + 54(61355) 4+ 36(60554) 
+ 14(55424) + 10(55343) + 10(54533) + 14(52544)} 
— 4.5°Q°E + 184. 5E* + 26-5°Q?E?U — 8-5QEU*. 
If now we were to multiply by U, and proceed as 
before, we should obtain the value of U*%; thence we 
might pass (Art. 29) to 2U°; and D, would be given by 
(Newton’s theorem) 
6D,=2D2-+-6D,D)-- 3b? = = 6°" or 
14.5 
D,=5°(2-58Q8-4 aE ~SU9. 
But it is not necessary to calculate this coefficient, since 
it has been already found (see Sec. II., Art. 27) that 
Dg= 0, 0,05 0, 0;0,=54Q°. 
44, The equation in @ is, therefore, 
+ 2-5 QEH + 2-57 Q'H? + 5° QE? @? — 5" (58 Q* — E*) EO 
+54O8—0, 
which coincides in every point with the result obtained by 
Mr. Cockle. (See § 12 of his “ Researches in the Higher 
Algebra,” published in the present volume.) 
45. The foregoing calculations are laborious, but they 
