4 Mr. \V. Brennand. [Dec. 11, 



From these forms for G and G,'!,,* a variety of others may be 

 obtained, an well as expressions in the form of integrals for spherical 

 harmonics of the second kind. Corresponding forms may also be 

 established for oblate spheroids. 



7. The expansion of the reciprocal of the distance between two 

 points plays an important part in the application of these investiga- 

 tions. It has therefore been found in ellipsoidal harmonics and 

 thence, by reduction, in harmonics of the spheroid, circular cylinder, 

 and paraboloid of revolution, and its application has been briefly 

 illustrated in finding the general term in the expansion of the 

 potential due to the magnetism induced in an ellipsoid placed in any 

 field of force, and in finding the electrical capacities of surfaces 

 inverted from ellipsoids. In the same connexion, I have also found 

 the expansion for the potential due to a thin shell bounded by similar 

 and similarly situated ellipsoids, the density of which varies inversely 

 as the cube of the distance from a fixed point. 



8. In the last part of the paper I have shown how to prove what 

 Heine terms " addition theorems " in the case of spheroidal harmonics r 

 and thence, by reduction, in the case of Bessel's functions. 



II. "Photometric Observations of the Sun and Sky." By 

 WILLIAM BRENNAND. Communicated by C. B. CLARKE, 

 F.R.S. Received October 30, 1890. 



(Abstract.) 



1. The paper begins with a short account of the various papers 

 communicated by Sir H. Roscoe, and published in the Transactions 

 of the Royal Society. 



2. My observations were made at Dacca, East Bengal, in 1861-66, 

 repeated at Milverton, in Somersetshire, during the last two years. 

 My first experiments were directed to ascertaining the action of the 

 sun on sensitised paper exposed at right angles to the solar rays for 

 different altitudes of the sun, and largely to ascertaining the laws of 

 distribution of the actinic power in the sky. 



I take no observations except when the sky is quite clear. 



3. The method of measurement I adopted is the darkening pro- 

 duced in sensitised paper. I cut strips from one uniform sheet of 

 ordinary photographic paper. My observations being relative, I 

 obtain the same results (ratios) with any paper. 1 compare ulti- 

 mately the effects of the sun and of a candle on this same paper. 



4. I assume that, in burning a stearine candle, the chemical action 

 is proportional to the material consumed ; I have taken as my unit (i) 



