Cambridge Philosophical Society. 909 



radiation ; and on the quantity of heat given out by the sun. Sug- 

 gestion of an improvement on the ice calorimeter." 



D. "Approximate places of seventy-six ruby- coloured, insulated 

 stars, noticed in the course of observation, in either hemisphere." 



E. " Geodesical determination of the site of the 20-foot reflector 

 at Feldhausen, with respect to the Royal Observatory at the Cape 

 of Good Hope." The co-ordinates of the site were ascertained to 

 be 17,595 feet in the direction of the meridian to the south, and 

 5190 feet in that of a parallel to the west, from the centre of transit 

 instrument at the Observatory, corresponding to 2' 53""55 of latitude 

 and 4 s- 1 1 of longitude. . 



It would be unpardonable to conclude our long extracts without 

 transcribing the sentence with which the author closes his laborious 

 undertaking. " The record of its site (i. e. of the great reflector) 

 is preserved on the spot by a granite column erected after our de- 

 parture by the kindness of friends, to whom, as to the locality itself, 

 and the colony, every member of my family had become, and will 

 remain, attached by a thousand grateful and pleasing recollections 

 of years spent in agreeable society, cheerful occupation, and unal- 

 loyed happiness." 



X L V 1 1 . Proceedings of Learn ed Societies. 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



[Continued from vol. xxxii. p. 144.] 



Bee. 6,/~\N the Critical Values of the sums of Periodic Series. 

 1847. yJ By G. G. Stokes, M.A., Fellow of Pembroke College, 

 Cambridge. 



There are a great many problems in heat, fluid motion, &c, the 

 solution of which requires the development of an arbitrary function 

 of x,f(x), between certain limits as o and a of x, by means of func- 

 tions of known form. The form of the expansion is determined, at 

 least in part, by the conditions to be satisfied at the limits ; and it 

 is usually considered that these conditions are satisfied by adopting 

 the form of expansion to which they lead. Thus, if the problem 

 requires that/(o) and /(a) vanish, it is considered that this condition 



is satisfied by developing f{x) in a series of sines of — and its mul- 

 ct 

 tiples. But since an arbitrary function admits of expansion in such 

 a series, the expanded function is not restricted to vanish at the 

 limits o and a. It becomes then a question, how shall we know 

 when the expanded function does really vanish at the limits, and if 

 it does not, how are such expansions to be treated, and are they of 

 any practical importance ? 



In considering the logic of such developments, the author was led 

 to perceive in what manner the evanescence of f(x) at the limits can 

 be ascertained, or else the values of/(o) and /(a) obtained, from the 

 development itself, even when the series cannot be summed, by ex- 



