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Proceedings of Royal Society of Edinburgh. [sess. 
positive and negative ; and placed at equal successive distances JA : 
so that we now have 
-^P(x,0)='2 ( -lyfx+ifoy . . (19), 
or, as we may write it, 
- Co = P(*, f x + iX ’ °) • • • • ( 1 9 ')> 
where 
D(.r, 0) = 0) - 4>(x+ o) (20). 
In (19), P denotes a space-periodic function, with A for its period. 
This formula, with t substituted for 0, represents - £*, being the 
elevation of the surface above undisturbed level at time t , in 
virtue of initial disturbance represented by (19). 
§ 14. Remark now that whatever function be represented by <j>, 
the formula for P in (19) implies that 
P(^ + A,0) = P(x,0) (21), 
which means that P is a space-periodic function with A for period. 
And (19) also implies that 
P(z + 1A,0)= -P(z, 0) (22); 
which includes (21). And with the actual function, (18), which 
we have chosen for <£(x, 0), the fact that <f>(x, 0 ) = <£(- x, 0) makes 
P(«, 0) = P( - x, 0) (23). 
Thus (19) has a graph of the character fig. 5, symmetrical on each 
Fig. 5. 
side of each maximum and minimum ordinate. The Fourier 
harmonic analysis of P(#, 0), when subject to (22) and (23), gives 
T» / A\ A 2 7 TX , 2 77"i3? a r2 7 TX / Ci A \ 
V(x, 0) = Aj cos A 3 cos 3 f A 5 cos 5— — • • • (24). 
AAA 
§ 15. Digression on periodic functions generated by addition of 
