456 Scientific Intelligence.  [Jone, 
Let 4,5 Yo Ys) Ys» Se. represent the successive values of 7; 
then we shall have by Taylor’s theorem 
a: ee 
y,= 4 3 tk. 
Ye = tet Te: Tat & 
yy, = 4230453 ay 
dy d7 y n® 10% 
Ye Rich geht Ge Tg tte 
But since by the theory of exponentials 
2 zx 
ea ltat+ pstpagt &. 
Therefore @— l=xt+ S+isct &e. 
and as « may be any function whatever, if we substitute for it suc- 
eessively 
dy dy dy 
sao ga 7 7; 3 ke. 
we shall obtain 
d 
mW yp eee ay ow 
€ ie w+ 73 t ke. 
few dy dy But 
e —1l= 7 20+ 53-3 t &. 
ay 
——sw dy dy 3 2? 
4 _ = — ———- 
¢ L=73v+75-+ Tat & 
dy 
Tbe met dy n? w? 
ée ee ee SS eae a a ees 
If now we transfer the exponents of the powers of dy to the 
characteristic d, the second members of the preceding equations 
will become 
dx wt dz? * tra + &e. 
ay ay 2? 4? 
an ee eet 
qe 24 + gs + Tet &. 
dy. d? 4 3 w? 
a34 + a+ 73 t & 
dy dy n> w? 
a" + oe is &e. 
And hence we may regard 
aan 
