Mr. W. H. Walenn on Unitation. 121 



of the value of n. If 



we shall have 



E 2 



4(p + r)' 



or the total heat in n lamps is independent of the number of 

 lamps. 



The heat generated in each lamp will then vary inversely 

 as the number of lamps. 



St. Louis, Dec. 30, 1879. 



XVI. On Unitation. — IX. Practical Remarks thereon, together 

 with Examples. By W. H, Walenn, Mem. Phys. Soc. 



[Continued from vol. v. p. 218.] 



37. Tf^HE distinction between a unitate and the remainder 

 JL to a division is shown in articles 24, 26, and 27 ; 

 and in articles 31, 33, 34, and 36 the unitation-formula 



V s x=a n (r-Z) n - l + a n - 1 (r-Sy- 2 + ... + a 3 (r-8y 



+ a 2 (r — $) + a i 



is compared with the ordinary formula expressing the scale of 

 notation of a given number N. In articles 29 and 30, the ab- 

 sence of a , as a symbol, from the formula 



U { N=a„(r-8) n -' + a„_,(»-S)"- 2 + . . .+«,(r-8)> 



+ a 1 (r-S)°+«_ 1 (r-S)- 1 + «_ 2 (V-S)- 2 +.. 



+ a_ (n _ 1) (r-S)-<* ! - 1, + a_„( J .-S)-" 



is explained, a being the decimal point ; a may, in general, 

 be taken to be the origin from which the order of the digits of 

 N is to be reckoned in either direction. 



The use of these results and comparisons is manifest through- 

 out the whole subject. The first step in their application is to 

 obtain easily the unitate to a given number. Although the 

 general unitation-formula will, in all cases, furnish this value 

 by substitution, there are short methods which point out dif- 

 ferent processes according to the algebraical form of 8 in U$N. 

 The practical work in some of these methods has been illus- 

 trated in the papers I., III., and V. ; but an analytical treat- 

 ment of the subject is possible in certain cases. Some of these 

 cases will now be examined. 



38. The algebraic form of U 5 N, which may be written 



