1176 
MR. R. F. GWTTHBR ON THE DIFFERENTIAL 
Then 
/(^ — X, TT — p, y.2, . . ) 
_ Jyd{d+l){d+Z)IZ'. ^-A'^+l)(f/+2)(^^+3)/8+^f(rf+3)/3 ^d{d^■\){d+^){d+Z)!i^-d{d+i)-d%d+Z)|Z ^-d 
X /(f - X, ,7 - P, Y,. . .).(3a). 
To put the multiplying factor in a more concise form, we will use Halphen’s 
nomenclature. If a term contains the differential coefficients y^, y^, . .we will call 
the number of differential coefficients in the term its degree, and write it dx, and the 
sum of the indices of differentiation its weight, and write it w. 
Considering the transformation to be made in the ultimate form of the equation, 
we see that 
7r — p = DX“^j'“^('7r — P) 
y-2 . *. . - 
2/3 ~ ff- &c. 
yj. = + &c. 
From the mode of formation of the determinant, the degree of the terms in each 
coefficient of ^ — x and tt — y) is uniform, and their weight is uniform, so that we 
may consider the weight and degree of each coefficient. 
Let dx and iv denote the degree and weight of the coefficient of the highest power 
of 77 — p in the expanded determinant. 
We thus determine the multiplying factor to be 
From this we conclude that 
d ~h dx — 
IV — 
d {cl + 1) ((^ + 2) 
3! 
cl (cl + 1) (d + 2) (3d + 5) d (d + 3) 
4! 2 
results which can be verified by arranging the terms in the determinant in descend¬ 
ing order and summing the quantities required. 
It is also easy to see that if dx ( m, n) and iv (m, n) are the degree and w^eight of 
the coefficient of (77 — pY~’^ — x)” 
dx {m. n) = dx m 
IV (yn, n) = tv — n, 
or the determinant is a homogeneous function of tt —p and of the differential 
