378 
PROF. LOUIS YESSOT KING ON THE CONVECTION OF 
It is not difficult to prove from (ll) and (13) that 
£V 003 ' de = 7rl 0 ( 2 ) and j^cos e e zcos ’ de = ttIj ( 2 ).(14) 
Making use of the relation( 8 ) I n+1 K„—I„K„ +1 = (l/z) cos 71tt for n = 0, we obtain 
finally Q = 2 Ajctt, and thus for the temperature at any point the expression 
0 = (Q/2™:) e^Ko (nr), .(15) 
which is H. A. Wilson’s( 9 ) solution for a line source. 
Let u (g) dg represent the total flux of heat from an elementary portion dg of the 
,-r-axis between x = 0 and x = /3 0 . The contribution of this element to the temperature 
at any point is given by (15) in the form 
d0 = (1/27nc) u (g) d^e n(x ~ i] K 0 1 + (x—g)*] \. 
Since equation (5) is linear the temperature due to a distribution of line-sources 
along the cc-axis between x = 0 and x = /3 0 is obtained by integration, 
rPo 
V) = (V 2 ™)^ u (f)e n(z_f) K 0 |ri v/[^ + (a;-^) 2 ]|^. . . . (16) 
The boundary condition over y = 0 is expressed by the relation 
2ttk0 (x) = j" u {%) e n{x ~ i) K 0 1 n (x—g) | dg. .(17) 
If the temperature is prescribed over the boundary, equation (17) constitutes an 
integral equation for the determination of u (^) ; if the flux of heat u (£) is prescribed 
over the boundary from x = 0 to x — /3 0 the same equation gives the temperature of 
the stream in contact with the boundary. In either case the total heat-loss of the 
cylinder per unit length is given by 
H = 
•00 
u {£) d£, 
Jo 
(18) 
and the temperature at any point by (16). 
Section 4. Calculation of Heat-Loss wider the Hypothesis of Constant 
Flux over the Boundary. 
The solution of the problem in hand which gives results in best agreement with 
experiment for the case of convection of heat from small cylinders is that obtained by 
( 8 ) Gray and Matthews, loc . cit ., p. 68. 
( 9 ) H. A. Wilson, “ On Convection of Heat,” ‘ Proc. Camb, Phil. Soc,,’ 12, p. 413, 1904. 
