202 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
28. Indication of a General Method .—If the coefficients a or A with suffixes exceeding 
10 were all zero, the expression for Z in the second middle wave could be transformed 
at once to the form 
/I 5 V [ Q 2 (<r+u) 
' (T 3 <j) [ <X 
+ K2 + Lg'U, 
or to the form 
+h+ , u . 
(T 
But if these coefficients do not vanish a transformation of the same kind is still possible. 
We have, for example, 
2 5 5 ! (o- 2 + 1 lu 2 ), 
2 7 5 ! u (3cr 2 -f 1 lu 2 ), 
and thus Z can be expressed either in the form 
(i + K 2 + L 2 u + M 2 (o- 2 + 1lu 2 ) + N,u (3 o- 2 +1 lu 2 ) + ..., 
or in the form 
(~. ) I ( ^———j + k 2 + l 2 u + m 2 (cr 2 + 1 lu 2 ) + n 2 u (3o- 2 + 1 lu 2 ) + ..., 
where the factors whose coefficients are written K 2 , L 2 , ... , or k 2 , l 2 , ... , are the 
homogeneous rational integral functions of u and u of degrees 0, 1, 2, 3, ... , which 
satisfy the differential equation for Z. 
When this transformation is effected we may proceed to determine the second reflected 
waves. The first stejD is to find sets of coefficients analogous to A 0 , ... , A 3 and 
a w ... , a b . The next is to find sets of coefficients analogous to Ifi, B_,, ..., determining 
the loci in the plane of (r, s) along which x 0 = 0 and x 0 — c during the time that these 
reflected waves are being generated. By means of the coefficients analogous to B l5 
B,,, ... , sets of coefficients analogous to A 6 , A 7 , ... , and a 6 , a-, ... , may be found, and 
thus the second reflected waves may be determined. 
From the formulae for Z in the second reflected waves that in the third middle wave 
may be found, in the same way as the formula for Z in the second middle wave was 
found from those in the first reflected waves. 
The method of solution can be continued, and gives a theoretically complete solution 
of the problem ; but when arithmetical computation is attempted, failure may arise 
through approximate equality of groups of terms with opposite signs, so that some 
quantity, which ought to be calculable to five figures, for example, may only be calcu- 
