On Systems of Rays. 
29 
o-2087 )saay | soe 
os ox" or owdy = du: Swdz.’ 
_ 80 &F 60 &V 6Q &V 
~ ba dvdy Sr by? ou dydz’ 
on MBY MEY | EY 
~ 86 dvdz br Bydz bu Bz?’ 
oO eV 6 &V Ps) V 
oF = Svd2’ * or Byda" a = =e 7 (4) 
387 3087 Baer 
~ 8a dwdy’ 3 dydy |) Sy Seay” 
oe 80 &V 82 &V 02 &V 
~ 8c dxde/ Sr Sydz’ bu S282’ 
80. 3087 (ASF AST 
J 
We have introduced, in the equations ( Y*), the terms A%8Q,...A78Q, that we may 
treat as independent the variations és, ér, dv, 6x, which are connected by the condition 
62 =0. 
To determine the multipliers \",...4%, we are to observe that in deducing the 
"By 8s Bzdy er Sydy Sy Seay | 
foregoing equations, the relation © =O between the four variables o, 7, v, x, has been 
supposed to have been so expressed, by the method mentioned in the second number, 
that the function Q when increased by unity becomes homogeneous of the first dimen- 
sion with respect to «, 7, v; in such a manner that we have identically, for all values 
of the four variables o, 7, v, x, 
so 8 ‘ 
os tty Bivigs —=2 115 (B*) 
and therefore, 
S07 [00 f ao adj 
Aaa aa Tee 
Bre) ‘ 0) x rQ _ 
¢ Oaédr ¢ or? ¥ Sreu” (C’) 
Siu cone Katee c 
so 2a | Sa _ 20 
q oodx +e oro Sudy ox” 
Hence, and from the conditions (/”*), relative to the homogeneity of the func- 
tion WV, it is easy to infer that the multipliers have the following values ; 
' , OV 
AM= —o; A= —73 AM = —v; AMHe's AMHe’s AMHo's AMHR LD: 
(D*‘) 
VOL. XVII. I 
