1894.] On Functions connected with Tesseral Harmonics. 45 



This is below the experimental value for round pipes, and is about 

 half what might be expected to be the experimental value for a flat 

 pipe, which leaves a margin to meet the other kinematical conditions 

 for steady mean-mean-motion. 



(o.) That the discriminating equation also affords a definite ex- 

 pression for the resistance, which proves that, with smooth fixed 

 boundaries, the conditions of dynamical similarity under any geo- 

 letrical similar circumstances depend only on the value of 



, 



fjf dx 



rhere I is one of the lateral dimensions of the pipe ; and that the 

 expression for this resistance is complex, but shows that above the 

 critical velocity the relative-mean-motion is limited, and that the 



jsistances increase as a power of the velocity higher than the first. 



III. '' On certain Functions connected with Tesseral Harmonics, 

 with Applications." By A. H. LEAHY, M.A., late Fellow of 

 Pembroke College, Cambridge, Professor of Mathematics 

 at Firth College, Sheffield. Communicated by Professor 

 W. M. HICKS, F.R.S. Received March 24, 1894. 



(Abstract.) 



The transformation of a zonal harmonic referred to a pole on a 

 sphere to another pole on the same sphere, and its expression in a 

 series containing the 2n + l harmonics of the same order referred 

 to this new pole, is an operation frequently employed in physical 

 research. The purpose of this paper is the investigation of certain 



motions of the angular distance between the'poles which occur when 

 a general tesseral harmonic is transformed from one pole and plane 

 to another pole and another plane of reference. If the coordinates 

 of any point on the sphere when referred to the first pole are /3' and 7' ; 

 denoting the colatitude, and 7' the longitude; and if the co- 

 srdinates of the same point when referred to the second pole are f>' and 



1 ; ' denoting the colatitude and q the longitude referred to a plane 



irough the two poles, it is shown that 



,W,H/) cos 7717' = cos m<*[\ u M o P (") + 22; j . P (",?'>") cos rq > 

 (. n + r\ ) 



-\- sin m-i . 2 ^ - - v mr P V> '') sin rq, 

 n + rl 



