Mr. W. H. Walenn on Unitation. 271 



the principle of the conservation of energy and by spectro- 

 scopic observations) would seem to be a highly practical one, 

 much needed in physics. By it all idea of the inconceivable 

 jar of the collision of infinitely hard molecules (or atoms) is 

 avoided. On account of the perfect elasticity, all motions 

 take place with complete smoothness, so that a perfect state of 

 mobile equilibrium is rendered possible in nature, which (with- 

 out due precautions) may well be competent to deceive the 

 senses into the idea that in what we call "space " all is in a 

 state of rest. 

 London, 1880. 



XXXIY. On Unitation. — X. Practical Remarks thereon, to- 

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

 [Continued from p. 128.] 

 40. fTlHE next algebraic forms of 8 that claim attention are 

 J- those comprised under the formula Ujfc w N, k being a 

 fundamental base such as 9 or 11, and n a whole number as a 

 multiplier. This set of unitates may be said to be the entry 

 to the domain which enables unitation to furnish easily the 

 remainders to divisors in general. 



41. In dissecting the expression U^N, it is observable that 

 the next less number exactly divisible by k is N — U&N, and 

 that U;fc ra N = UfcN + km, in which m is a function of U w (n — U&n). 

 Each of the multiples of k up to Jen may give a separate value 

 when unitated to the base n„ Let the multiples of k be <%, a 2 , 

 c/ 3 , . . . a n (so that k = ai, 2k = a 2 , Sk = a 3 , &c.) ; then the series 

 ~Un(a,i, a 2 , « 3 , . . . a n ) may be 1, 2, 3, 4, . . . n, or it may be other 

 values. If V n (a 1 , a 2 , a 3 , . . . a n ) = l, 2, 3, 4, . . . n, this is the 

 same sequence in which the multiples of Jc enter into U^N, 

 and m = U TC (N— UfcN) ; if ~U n (a ly a 2 , a 3 ,...a n ) gives other 

 values, means must be found to reduce them to the set of 

 values 1, 2, 3, 4, . . . n, if U& re N is to be used for obtaining the 

 remainders to the divisor under consideration. 



42. If k=9, then a 1} a 2 , a 3 ,... w = 9, 18, 27, 36, ... n; and 

 to test whether U^N is applicable when Jcn = 72, for instance, 

 it is necessary to obtain U 8 (9, 18, 27, 36, 45, 54, 63, 72). 

 This is found to be =1, 2, 3, 4, 5, 6, 7, ; therefore U^N is 

 applicable in this case in a direct manner. U in is not appli- 

 cable when kn = 27 ; for then U 3 (9, 18, 27) = 0, 0,0: this 

 result is also evident from the simple consideration that 3 is a 

 submultipleof 9. If U*»=45, U 5 (9, 18, 27, 36, 45) = 4, 3, 2, 1,0; 

 this can be reduced to the required sequence 1, 2, 3, 4, by 

 taking m= — U 5 (n — U9N), or obtaining the complement to 



U2 



