366 
MESSES. C. AND E. CHAMBEBS ON THE MATHEMATICAL EXPEESSION 
0 = 5 [sin tmz{ — a»+0„+y»)] = 5 [sin tmz( — a *)] +2 K 
whence 
whence 
whence 
whence 
whence 
2 m=r-l 
b t =~% [a m sin tmz] ; 
0=5 [cos{mz(— «„+&»+ 7*)] =5 [cos^ mz{ — a*)] + ^ A, 
m=0 y m=0 y 
2 m=r— 1 f 
A i=7£ o [ a m cos -m 2 ] ; 
0=5 [sin^m.z(— a ro +^ ro +yj]=5 [sin^mz(— a m )]+^B 1; 
ro=0 s' m=0 y 
c>m=r - 1 ^ 
B, =- 2 K sin - m 2 ] ; 
1 L » <7 J’ 
0=5 [cos2-m2(— a m +j3 m -t-y Bl )]=S [cos 2^-m2(— a m )]+^ A, 
m=0 ^ m=0 # 
A 2 =- £ [a m cos 2 - m 2 ] ; 
2 r rl 0 L m £ 1 
/, 
0=5 * [sin 2~mz(— a m + 0 ro + y m )] =5 [sin 2- g mz{— a!)]+^B 2 
B,, = - 5 [a_ sin 2 -m 2 ]; 
T m=0 L ^ J 
&C. 
&c. 
&c. 
whence 
whence 
0=5 [cos £ { mz{ — + 0 m + y m )] — 5 [cos £ ~ a mz { — a m )] + ^ A„ 
m-;0 y m = 0 j 
2 m=r-l ^ 
A<=- 5) [a m cos t - mz\ ; 
* rr- L ^ J 
0=5 [sin ^ { mz( - a m + /3 m + y m )] = 5 [sintf{ms(— a*)]+£B„ 
m=0 ^ m=0 v * 
B t —~ 5 [a m sin t ~ n mz\ 
1 r . L J 
5. In any special inquiry, having found by (8) the numerical values of « 0 , a x , « 2 , 
# 2 , &c., A 15 B„ A 2 , B 2 , & c., we may insert these in the equations (7), which it will now 
be desirable to consider the significance of. If our object was simply to find two 
periodical phenomena which would jointly satisfy the r observations, then this could 
be done with the same degree of closeness in an infinite variety of ways ; for we might 
give to the several terms of the right-hand members of (7) any arbitrary values consis- 
tently with their sum being equal to the left-hand member, and so long as the same 
coefficient is taken of the same value in all the equations (7). But although all the 
varieties would agree in giving the same value of the combined phenomena at any one 
