ON ANNUITIES. 197 



"be comprehended (if at all) by others from the 

 example. 



' 35 and 45 133 35 and 50 123 



40 and 45 129 40 and 50 120 



4 3 



3 3 



5) 12 5) 9 



2 2 



38 and 45 131 38 and 50 121 

 38 and 50 121 



10 



2 38 and 47 127 Ans. 



5) 20 



Before proceeding further, I shall describe the 

 notation of which I intend to make use. It was not 

 the practice of the earlier writers to invent any dis- 

 tinctive notation of different contingencies, the first 

 attempt at which is found in the work of Mr. Baily. 

 Here, however, it was not carried to the full extent, and 

 Mr. Milne endeavoured to organise a system which 

 should take in every case, in which he succeeded perfectly 

 as far as distinct representation of all the cases which 

 occur. His symbols, however, are complicated and 

 strange, though I am clearly of opinion that they are 

 much preferable to the attempt to dispense with 

 notation altogether. The new principle which the 

 notation I now propose involves, lies in the treatment 

 of terms of years certain as lives not subject to con- 

 tingencies. Thus, if AB represent an annuity on the 

 joint lives of A and B, meaning that it is to cease 

 when either A or B dies, then B may represent an 

 annuity on the joint term of t years and B's life, to 

 cease with the first which expires ; or what would be 

 o 3 



