36 Exponential and Logarithmic Theorems. 



9. That a partial temper or slight degree of hardness, gives 

 most power to small masses, but in permanency of power, hard 

 magnets, whether in large or small masses, are always superior. 



10. That after steel has been tempered to the highest degree 

 of hardness, annealing may or may not be beneficial, according 

 to circumstances included in following summary. 



a. For large and massive single straight magnets, the best 

 cast-steel, tempered as hard as possible, is to be preferred. 



h. For straight-bar compound magnets, the same quality of 

 steel and degree of hardness. 



c. For compound magnets of busk-plate, the best cast-steel, 

 tempered to the utmost in boiling linseed oil. 



d. For horse-shoe magnets, if in a single bar, cast-steel anneal- 

 ed from file hardness, in a bath of oil at 550°. 



e. For compound horse-shoe magnets, cast-steel annealed at 

 from 480° to 500°. 



/. For compass needles, if single and heavy, hard cast-steel ; 

 if light, whether single or compound, cast-steel annealed at from 

 500° to 550°. 



g. For very light needles, the best cast-steel, annealed in boil- 

 ing Unseed oil. 



Art. IV. — A new way of obtaining the Exponential and Log- 

 arithmic Theorems ; by Prof. Theodore Strong. 



We shall suppose that a and N denote any two indeterminate 



quantities, and shall put 5^ =y, the logarithms being ta- 



log.(l + a) 



ken to any arbitrary base ; then we shall evidently have N= 

 (l + a/, (1); for by taking the logarithms of both sides of (1) 

 to the same base as before, we have log.N=y.log.(l4-a)=log.N, 

 which is an identical equation, as it ought to be. 



If we expand the right member of (1) by the binomial the^ 



orem, we get N = 1 + ya -f- — t-^ — ^ + T2^ "^ 



2/(2/ -!)■(«/ -2).(2/- 3) 



-12 3 4 "* +&C., which being arranged according 



(a^ a^ tt* 

 «— 2"+'3"~T"l" 



