436 Mr. Murphy on the General Properties 



of anv term = x i the difference taken horizontally, and 



K — 2 m •^ 



in the odd rows (as the 2w+i"') the law be the same, except that 

 the multiplier is , then the «'" row will always be a repro- 



duction of the first. Example for five rows: 



*1, 4, 9, l6, 2d, &c. First row. 



5 13 25 £1 61^ 

 2 ' T' 2 ' ~2 ' y ' 

 12 34 68 VU 172 

 '6' ~6' 6 ' 6 ' "6~ 



28 58 100 154 220 



'6' '6' T' IT' 6"' 



6 24 54 96 150 



6 ' T ' IT ' 6 ' 6 ■■ 



&c. 

 &c. 

 &c. 

 &c. Fifth row ; same as first. 



The inver,se problem, viz. given the definite integral of a function 

 in different circumstances, to find the form of the function itself, 

 is susceptible of various solutions, as those circumstances are 

 themselves varied: thus, if D, be the definite integral of any 

 function of x from x = a - h ^ -\ to a = » + h ^ ~l, 

 Di from i=a-3hj-i to .i = « f 3 /i ,/ - 1 , 

 A from .r = a - 5 /( V - 1 to CI + bh J -\, 



'5 

 &c. 



then if we take the sum of the .series 



* Thus if Ai be the vertical, aud A the horizontal differences, and w, any term in tlie 

 lirst line, «2 the corresponding in the second, ^&c. 



then A, («,)=-. -y , A, (a,) = - ^— ^ • -y- , 



. , . n-l A.u, . , , -2 A.tts 



&c. &c. 



