First, thanks to those that helped me with my previous question about how to make a 555 emit a short, single, positive pulse upon power-up.

Now...my project also requires using the same 555 to give a short, single, negative pulse to trigger another 555 (which is in flip-flop configuration) to 'on'.

I thought a capacitor in series with the output of the first 555 to the trigger of the flip-flop 555 would give the latter a short positive pulse and then a negative pulse as the first 555's timing cycle finishes, and this positive-to-negative voltage change would flip the second 555 to 'on'. (This has been done in 2 instances previously in this book...so I'm hoping this is the solution).

But I tried it and my circuit doesn't work properly. The negative pulse to the flip-flop 555 should be short...maybe it's being held too long?

 

That should work. Post your schematic, showing all component values.

 

post your circuit , do you mean this method of triggering

 

Okay, here is the circuit I already made. It is a reaction timer which counts and displays thousandths of a second. There are three linked 4026 decade counters (only one is shown) driving three 7-segment LED displays (showing tenths, hundredths, and thousandths of a second, respectively).

Pressing the Start Delay button will start the timer a few seconds after pressing. Then pressing Stop Count as soon as possible will freeze the display to show elapsed time. The goal is to test your reflexes. Reset resets the display to 000.

Upon power-up, the display is counting, which requires pressing Stop and Reset, so this book suggests you find a way to automatically freeze the clock and reset to 000 upon startup - by using a single 555 to give a negative pulse to the Trigger pin of the middle (flip-flop) 555 and giving a positive pulse to the Reset pin on all three 4026s.

So I also made the following circuit that was suggested to me. By itself it works nicely to give a short, single pulse at power-up:

http://forum.allaboutcircuits.com/attachment.php?attachmentid=69949&d=1403279095

So I'm thinking that the solution is to connect the Output pin of the above circuit directly to the Reset pins of all three 4026s and to the Trigger of the flip-flop 555 with a capacitor in between (to hopefully give a negative pulse during the high-to-low transition). I tried it and it doesn't freeze the clock at power-up.

 

I thought a capacitor in series with the output of the first 555 to the trigger of the flip-flop 555 would give the latter a short positive pulse and then a negative pulse as the first 555's timing cycle finishes, and this positive-to-negative voltage change would flip the second 555 to 'on'. (This has been done in 2 instances previously in this book...so I'm hoping this is the solution).

But I tried it and my circuit doesn't work properly. The negative pulse to the flip-flop 555 should be short...maybe it's being held too long?

A capacitor alone in series with a square wave output is not the proper solution.

What you need is a high pass filter, which is also a differentiator.
Put a capacitor C in series with the output followed by a resistor R to ground. This will give you a positive going pulse on the rising edge of the square wave and a negative going pulse (also negative voltage) on the falling edge. Select the values of C and R in order to obtain the desired time constant t = R x C.

With the resistor to ground only the positive pulse will be effective on the next 555 circuit.

If you want to use the negative pulse, connect the resistor to Vcc instead of GND. This will bias the output of the high pass filter to Vcc instead of GND.