182 Proceedings of the Royal Society of Edinburgh. [Sess. 
can never have all its terms of like sign. Therefore, although the case of 
normal co-ordinates was excluded from consideration, it is necessary to 
inquire whether, in any of the vibratory systems under consideration, the 
fundamental assumption referred to in § 1 above may be impossible of 
fulfilment. Should this be so, it will not follow that equipartition of 
energy holds in such systems, but it will follow that the test applied 
becomes indiscriminative. 
4. The equations of motion are of the type 
4 ®pl4 ”1“ +••••+ Ctpp£ p +•*•*+ Olpn£n 5 
and the solution is of the type 
m 
4 = 2 sin K* + Or) , 
1 
where — n r 2 = a 11 -\-\ 2r a 21 -\-\ 3r a sl + . . . +A mr a ml , and the As are subject to 
the m - 1 relations 
K'g{ a il + ^2q a 21 + A 3g a 31 + • • • • + ^mq a ml) = a iq' + Kq a 2q' + * * ’ * +A mq^mq' • (1) 
obtained by giving q' values from 2 to m inclusive. The quantities 
A 2 . • ■ Am are the usual undetermined multipliers used in the integration, and 
the second suffix, r or q, which has values from 1 to m inclusive, merely 
indicates one of the m values which each A may have. 
Using the notation a gi = k lg a lg , and postulating observance of the third 
law of motion, so that k lg Ma = M v where M g is the mass associated with the 
co-ordinate £ g , we have 

Mpg dgpT 5 
•h g 
and the equation connecting the As becomes 
Forming the similar equation in g-f 1, multiplying it by \ q , q , and subtract- 
ing from it the above equation multiplied by \ q > , a+1 , we get 
m q'—l 
^•q'q^q', ki r (X rq A r> g-|_ i)0>i r ~ (A q’' A q'^)CL\q’ 4* (Ag', g+lA rg ~ X q t q X r q+ i)a rq , 
2 2 
m If 
+ 2 2+1 Kq - X q >qX r< q+ i )a 9 . r = 0 . 
S'+l "'12' 
Multiplying this equation by k lq - , and summing for all values of q' from 2 
to m, there results 
m m m q'—l 
k\r(Xrq ~ Ar > 1 g+i)(l 4" k lq ,X q r q X q t t q+ijdir -2'2 k\ q '(Xq' t q +\X rq VA.i+i) 4 ' 1 ’ 
2 2 
m m 
~ 2i 2 4r(Ag', -lA r2 — X q , q X r ; q+ l)d q 'r = 0 . 
2 q'+l 
2 
2 
