112 
NOTE IN ILLUSTRATION OF CERTAIN GENERAL THEOREMS. 
[577 
The vis-viva function T may be expressed in terms of the normal-variables and 
their derived functions; viz, it is easy to verify that we have 
' 1 sin 2 V 
\C 2 <r 2 c 2 <r 4 
sin 2 <r 
T=l 
(mi' 4- vv + ww') 2 
(u' 2 + v' 2 + to'-), 
where w denotes - - (an + ¡3v) and consequently tu denotes — - (au' + fiv'); introducing 
7 7 
herein differentials instead of derived functions, or writing 
<f> (du) = \ [Jpp ~ S1 c ^f) (udu 4- vdv + wdw) 2 
+ \ 8 * n a (du 2 + dv 2 + dw 2 ), 
where w, div denote — - (au + ¡3v), - - (adu 4- fidv) respectively; then </> (du) is the 
function thus denoted by Dr Lipschitz: and writing herein t — t 0 = 0, and thence u — 0, 
v = 0, w = 0, a = 0, the resulting value of <f) (du) is 
/o (du), = ^ (du 2 + dv 2 4- dw 2 ), 
where / 0 (du) is the function thus denoted by him; the corresponding value of f 0 (u) is 
= \ (u 2 4- v 2 + w 2 ). We have thus an illustration of his theorem that the function (f) (du) 
is such that we have identically 
</> (du) - {d V{/ 0 O)}} 2 = 2/Ju) W ~ № 
where m is a function of u, v independent of the differentials du, dv, the value in 
the present example is in fact m 2 = c 2 sin 2 a; or the identity is 
<f> (du) - [d V(/o «)} 2 = [/»W “ V(/»} 2 ]> 
in verification whereof observe that we have 
j n -c \ _ d fo( u ) _ udu + vdv + wdw 
V(/oW) “ 2 V(/>) = V(u 2 4- v 2 4- w 2 ) 
— — (udu 4- vdv 4- wdw) 2 . 
Ca 
The value of the left-hand side is thus 
sin 2 a 
viz. this is 
or, finally, it is 
which is right. 
2 4 (udu + vdv 4- wdw) 2 + \ 0 (du 2 4- dv 2 4- dw 2 ), 
C“(T O“* 
c sin a ^ ^ —1 ( uc i u v( i v + w diuyl ; 
2^.2 
(ra 
c 2 a 
c a~ 
2fpü) y°( du ^ ~ Vi/»*»}] 1