34 
MR. G. T. WALKER OX BOOMERANGS. 
Aj -j- sin lit + cos nt + sin cos ‘Int + F\ sin Znt + 44^ cos 3 rG 
sin nt + Co cos nt + Do sin ‘Int -|- Eo cos 2iit + F'o sin + C'o cos 3/G, 
A 3 + Bo sin nt + C 3 cos nt + Do sin 2nt + Eo cos 2at, 
in which A, B, C , . . are constants. 
On substituting the values of Fj, Fo, we find 
. khXj t ni 
Ai = ' ( W’? "r 
“-i 
^ j(i: + A') +' ^ 
Jr L\A:f /fo" 
1 _L a; 4 
'^'10 
+ 
3^_+J 2 V + . 
/fo' 
Ao-= 
XU r //»i,“ , Ail\ Ije _j_ / 3//h.f — 2///-d — 2/;td + 3//i 'd ^ .' L/Zo*' — '2m '.— '2m^^ + 3///^ ^ 
-P l \ '<^ 1 " 
/f [2/o- + /d , /o- + 21 ^ 
+ 
/f.," 
uU, 
\n 
rV^ — [2 {I.f + /g-) U" + {Ij'^ + 4^ + V + 4'/) ~ cos 6. 
Our equations may now be satisfied by the infinite series 
/3i sin nt + cos nt + 8^ sin 2at + . . 
z= tto d- jSo sin nt + Vo cos nt . 
IV = ag -f- ^3 sin nt + '/o cos nt + . . . 
^\■hidL are convergent, since the ratio of the coefficients of sin u + 1 nt and cos u + 1 nt 
to tliose of sin rnt and cos rut proves to be ultimately comparable with mr. 
If we adopt the notation 
u- 
V A a" 
+ ) 
.) - C. 1 j 
a' 
/f o' 
U' 
+ ( 
K-p + /fo 
4 //^ 
= Dob 
m 
/f 
/ ^ 
-( A + 
/ffiA 
.0 = 
4 
\/fp 
/fpy 
o 
1 
the non-eircular terms on the leftdiand sides become 
?;?ni fi- XUpB^ -f 2mping — 2X?rUiy, 
X -r-r 0 k\JW 
“ 2 /upin^ + — —;y, 
fC 
— X (/f^® fi- /c.o') i/Utd — mUn.o -f- iniv -fi XUo'Ri. 
