LAWS OF DECREMENT. 297 



If the new plane has resulted from a decrement on 

 the edge h ', by one row of molecules, the lines i 6, 

 f c, must also be to each other as m to n, and we 

 should then have 



sin. \/ icb : sin. \/ i'b c :: m : n. 



But if the new plane has resulted from a decrement 

 by unequal numbers of molecules in height and 

 breadth, the ratio of i b to i c, should be as p m to 

 q n ; the letter p representing the number of mole- 

 cules abstracted in the direction of the edge i k, and 

 q representing the number abstracted in the direction 

 of the edge i d. 



The ratio of p m : q w, may be expressed by the 



fraction of P m . which is evidently the product of 



qn 





' jf _ 1 We may therefore obtain the values of p 



q ' n 



and q, whatever may be the particular values of 



ff m and z , if we divide P m , which expresses the 

 q n n q n 



ratio of the edges of the defect by , which expresses 



the ratio of the corresponding edges of the molecules, 

 or, which is the same thing, of the corresponding 

 edges of the primary form. 



There are two methods by which this division may 

 be effected, 



Thejirst is by finding the absolute values of m and 

 w, and of p m and q n^ by means of the tables of na- 

 tural sines, &c. and then reducing those ratios to 

 their lowest denominations in whole numbers; and 

 after dividing the one fraction by the other, reducing 

 the quotient to its lowest denomination in whole 

 numbers. The quotient so reduced would express 

 the law of decrement. 



