128 Sir W, R. Hamilton on Robert Construction 



values. For this purpose, it has been thought convenient to 

 introduce eight auxiliary numbers, a . . .h, which can all be cal- 

 culated by square roots, and are defined with reference to the 

 recent equations (B), as follows : 



77" « 77* 



a = 1 + 2 tan <£, ; 6=4 cos 2</> 2 cot - ; c — 2 cos 2(/> 2 — cot - ; 



6?=sec2(/> 4 ; e=sec2cf) 5 ; f=2 cos 2 $; <7 = 2cos<£; A = 2cos^-- 

 or thus with reference to the earlier equations (A) : 



a -~7~> b ~f } ~J~> P~' *' 



J— r > 9— r > l ~ r > 



and respecting which it is to be observed that c, like the rest, is 

 positive, because it may be put under the form 



/14-2V5 /5 + 2\/5 



C= V --g V~ 5~~' 



and 14 — 2*/5>5 + 2*/5, because 9>4\/5, or 9 2 >4*5. With 

 these definitions, then, of the numbers a . . . h, and with the help 

 of the following among other identities, 



cos -^- sec ^- =2cos3(/> — 2cos2</> + 2 cos (j> — 1 

 = 2(2cos0-l)cos2(/>-l, 



I form without tables a system of values as below, the early 

 numbers of which have been computed to several decimals more 

 than are set down. 



r « s =5 a = 2-23606 79774 99789 6964 



72 a 



b<2 = S ^~25 hz= 3 ' 79998 36545 96345 0138 



19 



c 2 = — -6 c = 0-00404 29449 23565 7641 



^ ^=l-f-(2-c) 2 d= 2-23245 25898 01044 7849 



i e 2 = H-(5-2«)(d-l) 2 e=z 1-34230 90137 74792 5831 



p=(5-2ay + 2e-l f~ 1-62349 00759 24105 2470 



g*=if p= 1-80193 78878 99638 5912 



h 2 =2+g h= 1-94985 58633 65197 2049 



sm 



^2 - =hUf-\){g~\)- 2) = +0-00000 06134 49929 1683. 



