314 
Proceedings of Royal Society of Edinburgh. [sess. 
factor, substitute 
:1ttx 
'2ttx 
cos j— — \. l sin — , 
A A 
or e 
L f? 
(34). 
The alternative makes no difference in the summation 
r+oo 
ion I dr, 
because the sine term disappears for the same reason that the 
sine terms in (29) disappear because of (30). Thus (33) becomes 
27 TLX 
. . .(35); 
A <-t 
)es}/ 
put now J(z + lx) = lot ; whence 
+ LX) 
dx 
€ A 
.. = 2 dcr, and lx= - o- 2 - z. 
s/(z -f lx) 
• . . (36). 
Using these in (35) we may omit the instruction |R,S} because 
nothing imaginary remains in the formula : thus we find 
8J2 [* , 8J'2 , A / 
- v - dae a .e a = € a. - f - • J —. • vtt . ( 37 ), 
J - 00 A 2 t Tj 
^ A 
27 TJ /2 8 
=€ A 'ji. 
(38). 
The transition in (37) is made in virtue of Laplace’s celebrated 
discovery j 
§ 17. Equation (38) allows us readily to see how near to a curve 
of sines is the graph of P(x, 0), for any particular value of A jz . 
It shows that 
g 27J-Z 47 TZ ' 47 TZ 
Ai = — a j AJA 1 = J^ . e a ; A 5 /A 3 = x /-|.e a 
Suppose for example A = 4 z\ we have 
. (38). 
e _ X = e _7r — '043214 ; Ag/Aj = *02495 ; A 5 /A 8 =*03347 . (39). 
Thus we see that A 3 is about 1/40 of A 1 ; and A 5 , about g 1 ^ of A 3 . 
This is a fair approach to sinusoidality ; hut not quite near enough 
for our present purpose. Try next A = 2 z ; we have 
•043214; 
e~ 2 -=- 001867; A 3 /A 1 = ‘001078 . 
. (40). 
Thus A 3 is about a thousandth of A x ; and A 5 about 1 J x 10~ 6 of 
A l . This is a quite good enough approximation for our present 
purpose : A 5 is imperceptible in any of our calculations : A 3 is 
negligible, though perceptible if included in our calculations (which 
