420 Mr green, ON THE DETERMINATION OF THE 



we see that the function f will always be two degrees higher than F. 

 But since our formula become more complicated in proportion as the 

 degree of F is higher, it will be simpler to determine the differentials 

 of V, because for these differentials the degree of F and f is the same. 

 Let us therefore make 



, _ 1 dV _ r /o' (« — x) dx dy' d % 



~ 1 m' fir ~ J >rTi ' 



i,ia;-x'Y + iy-i/r + {z-zr + u''} — 



then this quantity naturally divides itself into two parts, such that 



A =xA' + A", 



, ,, /- p dx dy d% 

 where A' — -^r J '^ ;;rr\ , 



{{x -x'Y + {y- yj + [%-%)' + u^}~ 



and A"=-f~ 



x'p dx' dy' dx 



{{x-xy + {y-yy + {z-%y + u^~ 



By comparing these with the general formula (1), it is clear that 

 M — 1 = n' + 1, and consequently n = n + 2. In this way the expression 

 (28) gives 



which coincides with (34) by supposing F=f. 



The simplest case of the present theory is where y(a;', y', x') = l, and 

 then by No 11, we have 0o'= 1 and &„ = 1. when A is the quantity 

 required, and as the general series (29), No 11, then reduces itself to 

 its first term, we immediately obtain from the formula (30), the value 

 of A! following, 



* A= , — -- (the \ — 7 — (35), 



2 

 because in the present case H^, = 1, « = 3, and n = n' ^ 2. 



Again, the same general theory being applied to the value of A" 

 given above, we get 



