164 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
These are of the types (23), 24). 
(iii) Ye 
This gives, by (4), since y*x = 0, 
and may therefore be considered as of the type (24). It is important to note, how- 
ever, both here and in the cases immediately following, that the transformation on the 
value of w will not hold after the elementary solution has been integrated for the pur- — 
poses of the general problem in (41). 
i 1d il 
(iv). Oo = 5 ee 
_ Ne Lia le *x) 7 
OS ae Ge a 
a ee eae *x) | g 
aa Phd 
dy 2 
Cy ae (52) 
=@§4= a = eae r) 
ro) a 5 VE 
leads to 
aE @H 3 — a5 1 
= Qukers fareRs py ONS hy UE =) = 
a age St igge ) aa ee : 
a2 ak 3-a, ad 
oye ee ae 1 “2 2 ; , ; baa 
deay ae ¢ a Om 
= ~3)2% ¥% 
w (a ea VE 
and ze =z =z =0 
() ab Bina, gu ® Penge ime 
Gay: = oa geY © 24 dx® X 
which may be further decomposed into a { 
ac i d_, 
Yer Nini ipa Mo a ogee: 
of the form (26), and 
D _ he Oo %=0 gush di_s ( 
6 dy ’ ’ =z 3 dx p) 
the displacements corresponding to which vanish. | , 
