764 
MR. A. McAULAY ON THE MATHEMATICAL 
plate. Since the plate is thiq, we may suppose the faces of the cylinder large com¬ 
pared with the curved surface, and may, therefore, neglect the portion contributed to 
the integral of equation (27) by the curved surface compared with the rest of the 
integral. Now by equation (26) the part of the integral contributed by the face of 
the plate is zero. Hence putting i for UV we have, approximately , at any point of 
the plate 
S i (VBK0 - 2CHSK0) = 0.(28). 
Assuming B and C to be scalars, this may be put in the form 
S/K -1 0 = 2CB -1 S?HSK _1 © 
(29). 
Now assuming, which will be approximately —exactly at the boundary—true, that 
i is perpendicular to K, we have 
0 = - iSi® + KSK -1 0 + fKSfK -1 0. 
Hence by equation (2), § 81, 
7* (0 + iSi®) = (A - CH 2 ) (KSK" 1 © + fKSt'K -1 0) 
+ B (VKHSK -1 0 + VTKHS/K -1 0). 
Let us split this vector up into its components parallel to the three vectors K, i, and 
VKH. For this purpose notice that since K is perpendicular to i, 
VfKH = - KSi'H + fSKH 
iK = (VKH + fS/KH)/SiH. 
Thus 
(0 + fSf@) = K {(A - CH 2 ) SK- 1 © - BS/K" 1 ©SiH} 
+ iStK" 1 0{(A - CH 2 ) SfKH/SfH + BSKH} 
+ VKH {(A - CH 2 ) SiK" 1 ©/SfH + BSK" 1 0} . . (30) 
from which, by substituting for S/K _1 0 from equation (29), we obtain 
ot (© + i'Si‘0) = B -1 SK- T 0[KB {A - C(H 3 + 2SVH)} 
+ i2C {(A — CH 2 ) SiKH + BSKHS/H} 
+ VKH {2C (A - CH 2 ) + B 2 }].(31). 
96. This transformation is not likely to give us clearer ideas of what takes place 
when a stream of heat is made to flow in the plate which is large compared with the 
