1905-6.] Dr W. Peddle on Vibrating Systems. 133 
d 3 m 3 = kd 2 m 2 , k 4= 1 . This implies denial of the third law of 
motion with reference to m 3 . More generally, the condition 
suits with b l m 1 = Jc 1 b 2 m 2 , c 3 m 3 — k 2 c 1 in 1 , d 2 m 0 = k 3 d 3 m 3 , if k-Jc 2 k 3 = 1 . 
This implies denial of the law with reference to two masses at 
least, and asserts reality of the squares of the periods. Any other 
condition than k l k 2 k 3 = 1 makes two squares of periods imaginary. 
6. The solution of the equations of motion, given above, is 
expressed by 
Aj sin (n^t + cq) A : /q 
A 3 sin (?q£ + <q) 1 /q 
Afl= 
A 2 sin ( n 2 t + a 2 ) A 2 /q 
, A^ 2 =- 
A 2 sin (n 2 t + a 2 ) 1 /q 
A 3 sin(» 3 t + a 3 ) X 3 /* 3 
Ag sin (?? 3 £ + a 3 ) 1 fx 3 
A x sin + cq) 1 Aj 
1 Aj /q 
> 
II 
A 2 sin (?q£ + a 2 ) 1 A 2 
, A = 
1- ^ /^2 
j 
A 3 sin ( n 3 t + a 3 ) 1 A 3 
1 A 3 /q 
where - n x 2 = a Y + \^b 2 + /^Cg, - — a 1 + A 2 b 2 + g 2 c 3 , - n 3 2 = tq + 
\ 3 b 2 + /a 3 c 3 , and A 1} /q, etc., satisfy in pairs the equations 
H a i + b 2 X + C 3J“) = b i + a 2 X + > I / 3) 
g{a\ H" b 0 X + c 3 V) = iq + d 2 A -f- a 3 g , f 
If we write the values of £ 15 g 2 , $ 3 in the form 
= a\ A l sin {n-f + cq) + a 2 A 2 sin (w 2 £ + a 2 ) + a' 3 A 3 sin (n 3 t + a 3 ) , 
$ 2 = ^'jAj sin ( n Y t + aq) + b' 2 A 2 sin (w 2 ^ + a 2 ) + & 3 A 3 sin ( n 3 t + a 3 ) , 
£ 3 = c' 1 A 1 sin (?q£ + oq) + c 2 A 2 sin ( n 2 t + a 2 ) + c 3 A 3 sin (n 3 t + a 3 ) , 
we get, on integration over a period which is long relatively to the 
longest of the three fundamental periods, 
2 {£i 2 } = "WV + n 2 a' 2 A 2 + n 2 a 2 A 2 , 
2{^ 2 } = w 1 2 6 / 1 2 A 1 2 + n 2 b' 2 A 2 + n 2 b' 2 A 2 , 
2 {4 2 } = V c 'i 2A i 2 +w 2 V 2 2 A 2 2 + n 2 d 2 A 2 , 
where the brackets indicate time-averages, as before. Hence we 
have 
2{ m i£i 2 ~ m 24 2 } = (% ft/ i 2 “ m 2 b\ 2 )n^ A x 2 + (wqa'g 2 - m 2 b r 2 2 )n 2 2 A 2 2 + 
( m i tt 3 2 — m 2 & 3 2 )n 3 2 A 3 2 , . . . . (4) 
with two other similar expressions. 
7. From equations (3) we find that A is given by the cubic 
