48 MR. AIRY ON THE LAWS OF THE TIDES 



mean which would have been obtained if observations had been taken at infinitely 

 small equal intervals. The same remark applies in a stronger degree if the whole 

 duration be divided into twelve parts. 



Let us use the term phase for an angle proportional to the time which increases by 

 360° in a complete tide ; and let it be assumed that the height of the water can be 

 expressed by the following formula : 



Ao+Aj sin phase -fAg sin 2 phase +A3 sin 3 phase 4-A4 sin 4 phase 

 4-Bi cos phase +B2 cos 2 phase +B3 cos 3 phase +B4 cos 4 phase, 

 and suppose that the complete tide, or 360° of phase, is divided into sixteen equal 

 parts and the mean height in each part taken. 



The mean height in the first part will be, 



AQ+-Aircos — cos g j +2:^A2rcos 0— cos -^j +^A3rcos — cos -^ j +^A4rcos — cos -g- j 



+;^Bi(^sm g - sm Oj + 2^B2(^sm -g - Sin Oj +3^B3(^sm -g - sm j +^B4(^sin y - sm OJ . 

 The mean height in the second part will be, 

 Ao+-Ai(^cosg - COS -g-j +2^A2(^cos-g - cos-g-J +3^A3(^cos^ - ^os-g j +-A4(^cos-g - cos- 



, 8„ / . 2?r . ttN 8 /. 47r . 29r\ 8 „ / . Gtt . SttN 8 „ / . Stt . 4t\ 



+;;^i(«"^ T - «^^ sj +2;^^2(sin -g- - sin -^) +3^B3(^sin ^ - sin-g j +4^64 (^sm -g- - sin ^ j, 



and so on. 



Now if we group these in the following manner, 



(lst+5th+9th + 13th) + (2nd+6th + 10th + 14th) — (3rd+7th + llth + 15th) 



--(4th4-8th+12th-f 16th), 



32 

 the sum will be — A.. 



If we group them in the following manner, 



(lst+5th+9th + 13th) — (2nd+6th+10th + 14th) — (3rd+7th + llth+ 15th) 



+ f4th + 8th + 12th + 16th), 



32 

 the sum will be— B.. 



If we unite the adjacent means and group them thus, 



(1 st+2nd+9th + 10th)4-(3rd + 4th+llth + 12th) — (5th+6th + 13th+ 14th) 

 — (7th + 8th +r5th + 16th), 



32 

 the sum will be — A9. 



If we group them in this manner, 



(lst+2nd+9th+10th) — (3rd+4th+llth + 12th) — (5th+6th + 13th+l4th) 



+ (7th+8th + 15th + 16th), 



32 

 the sum will be — B,. 



