36 Professor Hamitton’s Third Supplement 
together with the conditions relative to Q, Q’, namely (B*), (C*), and the follow- 
ing, 
6Q! 6Q/ 6Q’ " 
' ‘ i = Oeil 
o 30’ 5 ace Sy’ uv Su! me ’ 
ea’ ea’ SQ’ 
‘ , ae , pe Ge = 0. 
2 oo” i do’ dr’ Ae 6a du’ ; 
eo! 82/ SQ! £ 
gta , ’ —— = (6) Xx‘ 
< 3a0u) Tin oe » Bev 3 ( ) 
Ps: ss: 
85 OU Spay id? eae ot 
OQ! OQ! OQ! 8Q! | 
© 08x ws or’ Ox tig ou'dx ae ; 
we have also the general equations 
OE iy 2S ON TOWRA, ' 
SM Hedaya (G’) 
by combining which with the foregoing conditions and with the partial differential 
equation (4°), we find the following, analogous to (4‘), 
so’ WV 8a’ BW 8a SW 
30 Sa® SY Seb * i Bwer’ 
80’ SW 8a SWB SW 
Jor Be'sy 1 Br B® | Ov by'dz”” 
80’ SV 8a SW 8a BW 
“Se Sve | Or By de By Be2 ? 
3a. 80° SW 8a SW SIV j 
ee = ae mm) 
sa 8a’ BW 8 Sw 8a OW 
Sr 7 8e Braz! SBD Soy’ | Ou erBe ? 
Sa 80° SW (a SWS OW 
dv 80" Suda" SF Sud | Sy’ Bude"? 
a _30' 30'S __ 30's ba’ BW 
Sx Sy 80" Syeu Be’ Byd/ TO dydF 
we 
and if we combine the conditions of homogeneity of the two functions W, 7, with 
the fundamental relation (7’) between these two functions, and with the properties 
of Q, 2%, and attend to (G'), we find the following expressions for the partial differ- 
ential coefficients of J, of the first order, 
