( 750 ) 
as for $ = 4. In the terms of order h * we may in the same 
way substitute other terms for the terms q x , ft, q x -, and ft 2 . 
dP, 
dP„ 
ft ft 
Then we have to reduce the terms — -— ft ft and 
oq s dq x 
To this end we introduce q\ and q' t in such a way, that 
q\ == + ~« 2 q ' ft + y ft ft ft 2 • 
q\ = ft -f —■ ft ft ft* + ft* ft • 
After these reductions it is evident that the terms of order A* in 
the first equation assume the form: 
j ~ ft 4* (ft—*,4-0 ^ J ft + c a —2*8^ ft + 
+ *, j t > + (j »,+*,) 2,* 2, - (a,-26,+e l)tl ( j 
4 
We now substitute y ft* for P 2 A s ft. This is allowed, because 
4 
substituting q t = Bh cos (nt-\- X) in — ft*, we obtain besides a term 
B*h*q t terms which are non-disturbing. 
We wish to investigate whether the disturbing terms in the two 
equations are again derivatives of the same function. For this we 
need not consider the terms with q x and ft*, in the first equation 
and those with ft and ft* in the second. The remaining terms become 
in the first equation : 
" (a * 24 *+«*>?*s.*+i( & .+1 «.)».•- 
In the second: 
«.+»,) 2 ,* - («,- 2 *.+«,) 2 , , 2 , + (b, + ~c. 
So finally we find that the disturbing terms are derivatives of the 
same function R; so the equations become: 
• , , dP ■ 
K + *'q*- = 
q* 4- » B — 
d ft' 
where R = Ph a q* - f Qh* q* -}- £7 4 , 
when P and Q are homogeneous quadratic functions of Vci x and 
|/« a and when U t is a homogeneous function of order four of ft and 
ft. The results found for the simple mechanism hold therefore for 
an arbitrary mechanism with two degrees of freedom. 
