358 REPORT— 1869. 



Section 12. — We now proceed to give certain formulae relative to the differ- 

 ential coefficients of the logarithms of functions 0, essentially the same func- 

 tions, be it remembered, as those which Jacobi writes with the letter Z. 



Schellbach's four fundamental equations are as follows : — 



r0,.r-re,7/=0/«,o'^(^l-±^ (i) 



r0.v-l"e^j=dy6,o\ff—f:i-) (2) 



l0..r-l"e..y=do'6.o'(L^-—\ (3) 



,~a,„.-r«,,=..= v(i,,-i,) (4) 



Of these, Schcllbach has fully proved the first. The second is immediately 

 derived from the iirst by putting x — kvi for x, and y — };il for y. The third 



may be found by putting in the second .r+ ^ — ^lifor .r, audy + ^ — ij/ifory. 

 "We thus obtain 



rdA--rdy=o./-{)/ If (y+ 1 -h'i\ -f(^c+ ^-i.A I 

 =do'e.ft'[-L- — ]. 



[ gy fjx- J 



Formula (4) may of course be demonstrated in the same way. 

 Putting y — hv'i for y in the second of these formula^ v/e have 



Z"ar-Z"0,y = «/fl,'r/'-l, ^-/r") (5) 



also we find 



Z"ftr-r0^y = «o'^O>-YJ.^ +<7,rA (G) 



7"«.»'-r'«j/ = «'r6)/r('/M-— J_~\ (7) 



Let .v=.v in these ftn'muhe, and wc have 



rf{,^ = %jrQ,y(fx~-^^^^ (8) 



rcjx^—do'ii.ft-icix-^l^ (9) 



r7ix=:~eo'e.jrf/ix-- \\ (lo) 



From those Schellbach deduces the following series of formula} by ea.sy 

 methods : — ■ 



