LAWS OF DECREMENT. 377 



The preceding parts of this section suppose the 

 angles known at which the secondary plane whose 

 law of decrement is required, inclines on one or 

 more of the primary planes. But it may sometimes 

 occur that the inclination of the secondary on the 

 primary planes cannot be directly obtained. In cer- 

 tain cases, where this happens, the laws of decrement 

 may be deduced from the inclination of the secondary 

 planes to each other. 



We shall suppose in the following examples of one 

 or two particular and simple cases of this nature, that 



the unit of comparison is expressed by , and the 



n 



ratio of the edges or other lines of the defect by 



P m . Whence will express the law of decre- 

 qn q 



ment by p molecules in breadth and q molecules in 

 height. 



It has been already stated that where an edge is 

 replaced by two similar planes, m will always be 



found to equal n^ and the fraction P y or its equi- 



q n 



valent , when reduced to its lowest terms in whole 



? 



numbers, will express the ratio of the edges of the 

 defect. 



1. Let us suppose the edge of a cube replaced by 

 2 similar planes as in mod. f, or the lateral 

 edge of a right square prism, as in mod. e. 

 And let the inclination of the secondary planes 

 to each other be called /. We shall find 



p_ _,.. R 



q ~ ~ tang. (| I&5) 



