1883.] Mr Hill, On functions of more than two valuables, etc. 313 



(5) On functions of more than two variables analogous to 

 Tesseral Harmonics. By M. J. M. Hill, M.A. 



[Abstract] 



A transformation analogous to the ordinary transformation from 

 rectangular to polar variables is applied to an equation similar 

 to that of Laplace, but containing i variables. 



A solution of the transformed equation is found which consists 

 of products of terms, each term being a function of one only of the 

 new variables. 



The solution thus found, omitting the first term, is called a 

 Normal Function, and is analogous to a Tesseral Harmonic. 



The Normal Functions possess a property analogous to the con- 

 jugate property of Tesseral Harmonics. 



The value of an integral corresponding to the integral of the 

 square of a Tesseral Harmonic over the surface of the sphere is 

 next evaluated. 



Then it is shewn that the solution of the transformed equation 

 is a rational integral homogeneous function of the original variables, 

 and that it includes under its form all the independent rational 

 integral homogeneous functions of the original variables of the 

 same degree which satisfy the original equation, but nothing else. 



The expansion of an arbitrary function of i — 1 variables in a 

 series of Normal Functions concludes the paper. 



(6) Observations of the Transit of Venus across the Sun, taken 

 near Kingston, Jamaica, at Cherry Garden, the residence of Oscar 

 Marescaux, Esq. Dec. 6, 1882 : by Dr Pearson. 



The Cambridge Philosophical Society will probably be glad to 

 receive an early account of the observations of the Transit taken 

 by one of its members. Their value will not be affected, though 

 the interest taken in them may perhaps be increased, by the fact 

 that they were made by one who belongs to the same college in 

 this University as the sole spectator of the first transit ever 

 observed, viz. that of Nov. 24 (o. s.), 1639. 



The Latitude of my place of observation was determined, 

 though from the nature of my instruments, not with very great 

 precision, by meridian altitudes of the Sun, Fomalhaut and 

 Achernar, the Pole-star not being available : and verified, in a 

 way, by my known distance from the Lighthouse to the east of 

 Port Royal, on the Palisadoes, at a place named Plumb Point : 

 which bore only a degree and a half east of South, at a distance 



