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PROFESSOlc A. R. FORSYTH ON THE 
and of n from 0 to y, the simultaneous zero values being excluded from the triple 
summations in the first line of the rightdiand side. 
o 
Proceeding in the same way with the expression for the increment of f, we find 
^fa_ey 
dt 
^ ^ P + l — m,y—n7)linn “P fa — 
I xw p A- a P 
I / I )H / ' n) <- = n + 1 O ya-l, ^-m,y-n£l,r,i+l,n 
d“ P-m, y-ii^l, m, ii + 1 H“ fa.-l, fi-m, y-mXh, m + 1, n 
"P fa.-!, p-m, y-n^!, m, n + 1 "P ^a-l, ^-TO, y-rXil, m+1, n} > 
1 
Imn S 
where again the summations are for all integer values of I from 0 to a, of m from 0 to 
/3, and of n from 0 to y, the simultaneous zero values being excluded from the triple 
summations SSS' in the first line of the right-hand side. 
Cj 
Effecting in these two results all the interchanges that correspond to the interchange 
of the variables u and v, we obtain the values of 
dt 
and also effecting in them all the interchanges that correspond to the interchange of 
the variables ?( and w, we obtain the values of 
dt ’ dt 
The expression for the increment of the derivatives of any function ^ (w, v, w), 
where (f) is unaltered in value by transformation, can he obtained in the same way as 
was that for the increment of tlie derivatives of a; it is found to he 
n ) y~'>Xlmn "P ^a — l,p + l — m,y — nVl'n>i ~h 7, ^-m,-y fl —nC/r.iJi} > 
with the same significance for as before. 
It thus appears that if the highest order of derivatives of the fundamental 
magnitudes that occur in a differential invariant be M, the highest order of derivatives 
of a function 6 that can occur is M + 1. 
6 . In order to avoid encumbering the memoir with vast masses of symbols, 
I projiose to exhibit the mode of constructing differential invariants up to the second 
order, that is, invariants involving derivatives of a single function (f) up to the second 
order and derivatives of a, h, c, f, g, h of the fir.st order. Then I propose to indicate 
what are the differential invariants up to the second order that involve more than 
a single function (f). And as the last part of the merely analytical portion of the 
memoir, I propose to state the results for differential invariants up to the third order 
l)ut to give practically none of the contributory analysis. Some idea of the protracted 
- V W'' i \ ) 
dt —UAw 
