1919 - 20 .] 
On a Class of Graduation Formulae, 
117 
Case I. — 0, i.e. j=l. 
Uq 
Table I. 
2to + 1 ( “° + “i + “-' 
^2 + ^-2 + 
-\-Um + U— m ). 
Case II. — k= 1, i.e. j = 2 or 3. 
No. of 
Terms in 
Formula 
=2/»+l. 
II 
Mg 
Po • 
Pi- 
Pt- 
Ps- 
Pi- 
Pi- 
Pi- 
Pi- 
Ps- 
Pi- 
Pit- 
Pn- 
Pl2- 
Pit- 
Pit- 
Pit- 
3 
1 
1 
5 
35 
17 
12 
-3 
7 
21 
7 
6 
3 
-2 
9 
231 
59 
54 
39 
14 
-21 
11 
429 
89 
84 
69 
44 
9 
-36 
13 
143 
25 
24 
21 
16 
9 
' 0 
-11 
15 
1105 
167 
162 
147 
122 
87 
42 
-13 
-78 
17 
323 
43 
42 
39 
34 
27 
18 
7 
- 6 
-21 
19 
2261 
269 
264 
249 
224 
189 
144 
89 
24 
-51 
-136 
21 
3059 
329 
324 
309 
284 
249 
204 
149 
84 
9 
-76 
-171 
23 
805 
79 
78 
75 
70 
63 
54 
43 
30 
15 
-2 
-21 
-42 
25 
5175 
467 
462 
447 
422 
387 
342 
287 
222 
147 
62 
-33 
-138 
-253 
27 
1305 
109 
108 
105 
100 
93 
84 
73 
60 
45 
28 
9 
-12 
-35 
- 60 
29 
8091 
629 
624 
609 
584 
549 
504 
449 
384 
309 
224 
129 
24 
-91 
-216 
-351 
31 
9889 
719 
714 
699 
674 
639 
594 
539 
474 
399 
314 
219 
114 
-1 
-126 
-261 
-406 
Case III. — k= 2, i.e.,;'=4 or 5. 
No. of 
Terms in 
Formula. 
M. 
Po* 
Pt- 
Pi- 
Vi- 
I 
Pt- 
Pi- 
Pa- 
Pt- 
Pit- 
Pn- 
‘Pi f 
Pit- 
Pu- 
Pit- 
5 
1 
1 
7 
231 
131 
75 
-30 
5 
9 
429 
179 
135 
30 
-55 
15 
11 
429 
143 
120 
60 
-10 
-45 
18 
13 
2431 
677 
600 
390 
no 
-135 
- 198 
110 
15 
46189 
11063 
10125 
7500 
3755 
-165 
-2937 
-2860 
2145 
17 
4199 
883 
825 
660 
415 
135 
-117 
- 260 
-195 
195 
19 
7429 
1393 
1320 
1110 
790 
405 
IS 
- 290 
-420 
-255 
340 
21 
260015 
44003 
42120 
36660 
28190 
17655 
6378 
-3940 
- 11220 
- 13005 
-6460 
11628 
23 
2185 
337 
325 
290 
235 
165 
87 
10 
-55 
-95 
-95 
-38 
95 
25 
30015 
42531 
4125 
3750 
3155 
2385 
1503 
590 
-255 
-915 
-1255 
-1122 
-345 
1265 
27 
930465 
121943 
118800 
109560 
94790 
75435 
528 18 
28640 
4980 
- 15705 
- 30580 
- 36432 
- 29670 
-6325 
37950 
29 
445005 
54251 1 
53040 
49470 43730 
36135 
27126 
17270 
7260 
-2085 
- 9820 
- 14874 
- 16050 
- 12025 
-1350 
17550 
1 31 
29667 
3381 1 
3315! 
3120 
2805 
2385 
1881 
1320 
735 
165 
-345 
-744 
-975 
-975 
-675 
0 
1131 
Case IV. — k= 3, i.e. j = 6 or 7. 
No. of 
Terms in 
Formula. 
M. 
Pa- 
Pi- 
Pt- 
Pi- 
Px- 
Pt- 
Pt- 
Pi- 
Ps- 
Pi- 
Pit- 
7 
1 
1 
9 
1287 
797 
392 
-196 
56 
-7 
11 
2431 
1157 
784 
28 
-308 
161 
-28 
13 
46189 
18063 
14000 
4550 
-3500 
-3605 
3388 
-770 
15 
138567 
46189 
38500 
19250 
-1540 
- 11935 
-5236 
10010 
-2860 
17 
96577 
28109 
24500 
15050 
3500 
- 5215 
-6916 
-910 
6500 
-2275 
19 
37145 
9605 
* 8624 
5978 
2492 
-679 
-2380 
-1918 
344 
2261 
-952 
21 
334305 
77821 
71344 
53508 
28812 
3801 
- 14364 
- 19908 
-11016 
7021 
18088 
-9044 
Case V. — k= 
4, i. e. j = 
8 or 9. 
No. of 
Terms in 
Formula. 
M. 
Po- 
Pi- 
Pt- 
Pi- 
Pi- 
Pt- 
Pt- 
*• ! 
Pt- 
Ps- 
P.o- 
9 
1 
1 
i 
11 
46189 
30313 
13230 
-7560 
2835 
- 630 
63 
13 
4199 
2183 
1350 
-135 
-465 
360 
-117 
15 
15 
96577 
42109 
30375 
5625 
-9885 
-3870 
8109 
-3705 
585 
17 
7429 
2803 
2205 
819 
-399 
-630 
63 
525 
-333 1 
63 
19 
22287 
7429 
6174 
3087 
-147 
- 1764 
-1071 
777 
12U6 1 
-1071 
238 1 
21 
3231615 
966073 
833490 
493920 
97755 
- 178290 
- 219177 
- 48930 
148410 1 
123165 
- 158270 
40698 
